New Techniques in Resolution of Singularities (Oberwolfach Seminars, 50) 3031321146, 9783031321146

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Oberwolfach Seminars 50

New Techniques in Resolution of Singularities Dan Abramovich Anne Frühbis-Krüger Michael Temkin Jarosław Włodarczyk

Oberwolfach Seminars Volume 50

The workshops organized by the Mathematisches Forschungsinstitut Oberwolfach are intended to introduce students and young mathematicians to current fields of research. By means of these well-organized seminars, also scientists from other fields will be introduced to new mathematical ideas. The publication of these workshops in the series Oberwolfach Seminars (formerly DMV seminar) makes the material available to an even larger audience.

Dan Abramovich • Anne Frühbis-Krüger • Michael Temkin • Jarosław Włodarczyk

New Techniques in Resolution of Singularities

Dan Abramovich Department of Mathematics Brown University Providence, RI, USA

Anne Frühbis-Krüger Institut für Mathematik Universität Oldenburg Oldenburg, Germany

Michael Temkin Einstein Institute of Mathematics Hebrew University of Jerusalem Jerusalem, Israel

Jarosław Włodarczyk Department of Mathematics Purdue University West Lafayette West Lafayette, IN, USA

ISSN 1661-237X ISSN 2296-5041 (electronic) Oberwolfach Seminars ISBN 978-3-031-32114-6 ISBN 978-3-031-32115-3 (eBook) https://doi.org/10.1007/978-3-031-32115-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Introduction

Rationale and Goals Resolution of singularities is notorious as a difficult topic within algebraic geometry, and people who master it have a “superhuman” aura. We hope that readers of this volume will gain some of the requisite “superpowers", in two closely related and intertwined ways. • One line of recent progress in resolution of singularities is algorithmic implementation, represented by the evolving work [11] and the introductory text [10]. • The recent work [3, 4] infused the subject with logarithmic geometry and algebraic stacks, two techniques essential for the current theory of moduli spaces. As a byproduct, the paper [2] was produced, see also [16], providing a short, simple and efficient functorial resolution procedure in characteristic 0 using just algebraic stacks. It should be said that in many circles algebraic stacks and logarithmic structures are also considered difficult, and many senior mathematicians feel under-qualified to impart the theories on their students. These two lines above are intertwined through the work [14, 15]. The stack theoretic and logarithmic algorithms have implications in resolution problems where these tools are indispensable, namely logarithmic and desingularization of families of varieties (see [3, 4]). The goals of this book are to introduce readers to (1) (2) (3) (4) (5) (6)

Algebraic stacks Logarithmic geometry Classical algorithmic resolution Logarithmic resolution Stack-theoretic weighted resolution Resolution in families and semistable reduction methods

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Chapter Synopsis A Computational View on Hironaka’s Resolution of Singularities (FRÜHBIS-KRÜGER) Knowing H. Hironaka’s famous result on resolution of singularities in characteristic zero [12] is just a first step toward actually determining a desingularization for a given variety or scheme. It took more than a quarter of a century, until E. Bierstone and P. Milman [5] and independently O. Villamayor [19] provided fully constructive proofs of the existence. But these constructions involve huge amounts of data, and even in reasonably small examples, calculations quickly get out of hands. So why not teach a computer to do these constructions for us? This was considered unfeasible, until G. Bodnar and J. Schicho presented a proof-of-concept implementation [6] in maple and independently A. FrühbisKrüger and G. Pfister provided one which became part of the SINGULAR-distribution. Central to Hironaka’s approach is a sequence of blow-ups at suitably chosen centers; crucial for the choice of the blow-up centers is a descending induction on the ambient dimension. These general ideas will be motivated and discussed before laying algorithmic foundations by a brief introduction to Groebner bases and their applications. The main part of the chapter is then devoted to an in-depth discussion of all theoretical and algorithmic constructions in Hironaka’s approach. Of course, blow-ups and their computation figure prominently in the discussion. The choice of center, on the other hand, is the very heart of the algorithm, and all of its building blocks are covered from theory to computational aspects, in particular order reduction, hypersurfaces of maximal contact and the role of exceptional divisors. All concepts are illustrated by many examples, accompanied by computational remarks and some sample computations with OSCAR [17], a currently evolving computer algebra system relying on the strengths of well-known specialized systems including SINGULAR, GAP and Polymake.

Stacks for Everyone Who Cares About Varieties and Singularities (ABRAMOVICH) What are stacks? We start by contemplating what varieties are, in concrete terms. It turns out that in such terms algebraic stacks are not very different! We then circle back to the world in which stacks were naturally introduced, namely moduli spaces: thinking of moduli, or classification problems, we encode them as categories fibered in groupoids. In order to glue objects, we endow these with a topology, arriving at the notion of stack. Finally to link them back to varieties, we introduce algebraic stacks. Our guiding examples for these are quotients by finite groups and moduli of curves. We then lightly discuss some basic constructions on algebraic stacks: coarse moduli spaces, tying stacks with schemes; root constructions, creating stack structure along a

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smooth divisor that makes classical branched covers canonical; destackification, which allows us to return from the magical world of stacks to the concrete world of schemes— this is absolutely essential when we obtain a stack-theoretic resolution, but applications demand a scheme theoretic resolution; and a beautiful description, due to Behrend, of quotients by groupoids. We also briefly touch upon stack-theoretic .Proj constructions and weighted blowups, which are treated more thoroughly in later chapters.

Introduction to Logarithmic Geometry (T EMKIN) This chapter provides an introduction to logarithmic geometry with an eye toward applications in resolution of singularities. Log geometry was introduced by Kato in 1988 following definitions and ideas of Fontaine-Illusie, and we start with a brief overview of the history, original motivations and numerous precursors of the theory that were existing before, including the usage of snc divisors and boundaries in resolution of singularities. Then we move on to the basic theories of logarithmic structures, logarithmic schemes and their morphisms. In particular, we study charts of logarithmic schemes and morphisms, modules of logarithmic differentials and the classes of logarithmically smooth morphisms and logarithmic blowing ups. We finish the chapter with a section containing a more advanced material—Olsson’s stacks .Log classifying log structures and their use to reinterpret (or introduce) various basic notions of logarithmic geometry, such as logarithmic differentials and logarithmic smoothness and regularity. This section will only be used in our sequel study of resolution of morphisms.

Birational Geometry Using Weighted Blowing Up (ABRAMOVICH, TEMKIN, WŁODARCZYK) This is a gentle introduction to resolution using weighted blowups, focused on examples. We review how, unlike the case of curves, classical methods cannot provide a resolution algorithm where in each step the worst singularities are modified. We then explain the main ideas of weighted resolution: invariants, centers, weighted blowing ups, coefficient ideals, through examples. We proceed to give a general structure of the proof and its elements. We close by discussing briefly how classical and weighted blowing ups fit into birational geometry, and a few questions that arise naturally.

Relative and Logarithmic Resolution of Singularities (TEMKIN) Our original plan for this chapter was to cover logarithmic resolution methods, including the case of morphisms. However, in order to provide a unified approach to (nearly) all functorial methods known so far and stress analogies, differences and flows of ideas

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between different methods, we decided to extend this plan and provide sections devoted to other methods too. This makes this chapter rather central in the volume, but also leads to a certain intersection with the material of other chapters, though we tried to make the exposition in non-logarithmic sections shorter and refer to parallel sections for details. So, what does this chapter contain? We start with describing general principles of constructing various desingularization and principalization methods. In particular, we emphasize the rather surprising observation, that (in all our cases) a very important part of such a construction is a careful choice of basic tools one uses called the framework of the method—the category of objects, the notion of smoothness, the class of blowing ups, the notion/formalism of ideals, etc. Then we consider the classical algorithm, already known to the reader from Chapter “A computational View on Hironaka’s Resolution of Singularities”, as an illustration of this general line of thinking. In particular, we re-interpret some of its ingredients in a way which fits generalizations/modifications one does further in order to obtain the new types of desingularization and principalization methods. After that, we describe the logarithmic methods in the case of logarithmic varieties over a field. First, we show how far one can advance without using stacks (or more or less equivalent machinery), and then we introduce stacks and relevant formalism of Kummer ideals and blowing ups to fill in the gap. In the next section, we show how these methods are extended to relative desingularization, which leads to functorial semistable reduction theorems. After that, we observe that Kummer blowing ups are a particular case of weighted blowing ups and switch to studying weighted desingularization and principalization methods, which are allowed to perform arbitrary weighted blowing ups at smooth (or .Q-smooth) centers. For simplicity, we start with the non-logarithmic case, which is also constructed with all details and proofs in Chapter “Weighted Resolution of Singularities: a Rees Algebra Approach”. However, the constructions there use cobordant blowing ups, and the constructions in Chapter “Relative and Logarithmic Resolution of Singularities” are based on stack-theoretic blowing ups, so the reader can see two different technical implementations of essentially the same method. After outlining the non-logarithmic case, we describe the modifications one has to perform in order to obtain weighted logarithmic desingularization. We finish this chapter with a section containing a brief overview of desingularization in broader categories—analytic spaces, formal schemes, quasi-excellent schemes. This is the most speculative section, as together with outline of known results it also discusses some expectations which were not checked yet in the literature.

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Weighted Resolution of Singularities (WŁODARCZYK) The purpose of this chapter is to discuss weighted resolution method in characteristic zero and its further generalizations and extensions. The method of weighted centers in characteristic zero are considered in the context of embedded resolution algorithms— these were originated by the fundamental work or Hironaka and developed and simplified in the work of Bierstone-Milman, Villamayor, Encinas-Hauser, Włodarczyk, Kollár [5,7,8,12,13,19,20] and many others presented in this volume in [9]. It is also a part of our recent program on embedded resolution algorithms of varieties and morphisms. One of the key idea of our work on the functorial logarithmic resolution of morphisms was to use more general centers. This approach was pursued in [3] and independently discovered by McQuillan in [16] and led to the weighted resolution of singularities. The algorithm in [3] is constructed in the language of the stacks and the centers written as .Q-ideals. It gives a fast resolution, which relies solely on a simple geometric invariant. In the subsequent paper of Quek [18], the method from [2] was applied in the context of the logarithmic resolution of singularities. In the paper [21], one considers this approach from a different perspective with emphasis on practical computations and further theoretical implementations in particular to positive and mixed characteristic. While the original method from [2] relies on the stack-theoretic blow-ups and .Q-ideals, in [21] we look at the weighted blow-ups from the perspective of the torus actions and rational Rees algebras. The operations introduced in Sections 2 and 3, of weighted cobordant blow-up, is presented in the language of the schemes with torus actions. Formally speaking, it is a particularly simple realization of the ordinary weighted blow-up .Y → X by a regular scheme B with induced action of a 1-dimensional torus .Gm . The geometric quotient is .B+ /Gm , so the space of orbits of its open subset .B+ ⊂ B is isomorphic to Y , but the stack-theoretic quotient .[B+ /Gm ] represents the stack theoretic weighted blow-ups. This language carries an important additional information in terms of the action, and the space B where all the transformations are carried. Another principal difference between the approaches is the presentation of the centers of the algorithm in [21] in the language of rational Rees algebras (see Section 4). The idea of rational Rees algebras, first introduced in the early posted versions of [2], is based upon a simple observation that by enlarging the gradation of a standard .Z≥0 -graded Rees algebra associated with the blow-up and thus passing to the finer rational gradation we often obtain a much nicer representation of the algebra. This approach allows to run the whole algorithm inside of the algebra of the center, which is itself defined by a fast recursive procedure. As the result, the construction requires a minimal number of steps, and what is important and somewhat surprising, the constructed center is automatically unique. The automatic uniqueness is even more relevant in the context of stacks, topologies, logarithmic structures and others where gluing of local charts can become cumbersome.

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Additional benefit of the approach in [21] is that it addresses the case of resolution and principalization with simple normal crossing exceptional divisors, which is critical for many applications. In particular, the computation of the coefficient of the exceptional simple normal crossing divisor of the resolution is an important tool in birational geometry. The algorithm is illustrated by multiple examples throughout the Section 4. The weighted characteristic zero algorithm cannot be applied directly in positive characteristic. In Section 5, we give several examples showing that the behavior of the centers and the invariant is difficult to control by the standard characteristic zero method. In the same Section 5, we show some applications of the ideas to resolution in positive and free characteristic. The method of cobordant blow-ups with weighted centers can be applied to certain classes of singularities including, in particular, to the isolated singularities whose generalized weighted tangent cone also has an isolated singularity, and others. In Section 6 we discuss the notion of Cox rings of the proper birational morphisms of any normal noetherian schemes introduced in [22]. The main idea is to give a nice representation of any birational proper morphism by a torus actions, generalizing the ideas of McQuillan [16] and Abramovich-Temkin-Włodarczyk [2] of the weighted resolution, and Abramovich–Quek [1] of the multiple weighted resolutions. Subsequently, we give some applications of the method to instantaneous resolution of singularities over a field of any characteristic.

Open Problems The theory of resolution of singularities is replete with open problems. We focus our attention on problems naturally connected to the rest of this volume. We open with questions of implementation of the various resolution algorithms introduced here. Related to these is the task of completing an implementation of destackification, and optimizing the algorithm in various ways. We continue with an extensive list of theoretical questions that beg to be addressed about improvements and generalizations of the results presented in this volume. Department of Mathematics, Brown University, Providence, RI, USA Department of Mathematics, Universität Oldenburg, Oldenburg, Germany Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel Department of Mathematics, Purdue University, West Lafayette, IN, USA

Dan Abramovich Anne Frühbis-Krüger Michael Temkin

Jarosław Włodarczyk

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References 1. D. Abramovich, M.H. Quek, Logarithmic resolution via multi-weighted blow-ups (2021) 2. Abramovich, M. Temkin, J. Włodarczyk, Functorial embedded resolution via weighted blowings up. arXiv e-prints (2019), arXiv:1906.07106 3. D. Abramovich, M. Temkin, J. Włodarczyk, Principalization of ideals on toroidal orbifolds. J. Eur. Math. Soc. 22(12), 3805–3866 (2020). MR4176781 4. D. Abramovich, M. Temkin, J. Włodarczyk, Relative desingularization and principalization of ideals. arXiv e-prints (2020). arXiv:2003.03659 5. E. Bierstone, P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997). MR1440306 (98e:14010) 6. G. Bodnár, J. Schicho, A computer program for the resolution of singularities, in Resolution of Singularities (Obergurgl, 1997). Progr. Math., vol. 181 (Birkhäuser, Basel, 2000), pp. 231–238. MR1748621 7. S. Encinas, H. Hauser, Strong resolution of singularities in characteristic zero. Comment. Math. Helv. 77(4), 821–845 (2002). MR1949115 8. S. Encinas, O. Villamayor, A new proof of desingularization over fields of characteristic zero, in Proceedings of the International Conference on Algebraic Geometry and Singularities (Spanish) (Sevilla, 2001), vol. 19 (2003), pp. 339–353. MR2023188 9. A. Fruhbis-Krüger, A computational view on Hironaka’s resolution of singularities (2023), this volume 10. A. Frühbis-Krüger, G. Pfister, Algorithmic resolution of singularities, in Singularities and Computer Algebra. London Math. Soc. Lecture Note Ser., vol. 324 (Cambridge University Press, Cambridge, 2006), pp. 157–183. MR2228230 11. A. Frühbis-Krüger, G. Pfister, RESOLVE.LIB: a SINGULAR 4-1-2 library for computing resolution of singularities (2002–2014). http://www.singular.uni-kl.de/ 12. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964); ibid. (2) 79, 205–326 (1964). MR0199184 13. J. Kollár, Lectures on Resolution of Singularities. Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, 2007). MR2289519 (2008f:14026) 14. J. Lee, Algorithmic resolution via weighted blowings up (2020). https://arxiv.org/abs/ 2008.02169 15. J. Lee, A. Frühbis Krüger, D. Abramovich, RESWEIGHTED: A SINGULAR 4-1-2 library for computing weighted resolution of singularities (2020) 16. M. McQuillan, Very functorial, very fast, and very easy resolution of singularities. Geom. Funct. Anal. 30(3) (2020), 858–909. MR4135673 17. The OSCAR Development Team, OSCAR — Open Source Computer Algebra Research system (2022). https://oscar.computeralgebra.de

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18. M.H. Quek, Logarithmic resolution via weighted toroidal blow-ups. Algebr. Geom. 9(3), 311–363 (2022). MR4436684 19. O. Villamayor, Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989). MR985852 20. J. Włodarczyk, Simple Hironaka resolution in characteristic zero. J. Am. Math. Soc. 18(4), 779–822 (2005) (electronic). MR2163383 21. J. Włodarczyk, Functorial resolution by torus actions (2022). https://arxiv.org/abs/ 2203.03090 22. J. Włodarczyk, Cox rings of morphisms and resolution of singularities (2023). Preprint arXiv:2301.12452

Contents

A Computational View on Hironaka’s Resolution of Singularities . . . . . . . . . . . . . . . . Anne Frühbis-Krüger 1 The Task of Resolving Singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Types of Resolution Tasks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Philosophy of Hironaka-Style Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computational Basics on Groebner Bases and Ideal Operations . . . . . . . . . . . . . . . . . . 2.1 Groebner Bases in a Nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Applications: From Elimination to Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Blowing Up: From Definition to Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Blowing Up at an Ideal Sheaf I on a Scheme W . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Blowing Up at a Regular Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Weighted Blowing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Locus of Maximal Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Order of an Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Simplest Case: X ⊂ An . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 General Case I: Local Situation at p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 General Case II: Covering an Affine Chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Maximal Contact and Coefficient Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Motivation: A Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hypersurfaces of Maximal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Coefficient Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exceptional Divisors and Blow-Up History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Motivation by Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Full Resolution Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Stacks for Everyone Who Cares About Varieties and Singularities . . . . . . . . . . . . . . . Dan Abramovich 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Statement of Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Prerequisites, Readings, Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Varieties and Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 What Is a Variety? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 What Is a Stack? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 What Is the Difference? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Categories for Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Moduli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Moduli of Curves as a Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Variety Is a Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Categories Fibered in Groupoids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Arrows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Quotients of Free Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Quotients in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Descent for Morphisms of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Descent for Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Stacks in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Algebraic Stacks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Representability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Smooth and étale Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Algebraic Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Smoothness of Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Quotients and Coarse Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Root Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Stack Theoretic Proj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Weighted Projective Stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Blowups and Weighted Blowups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Destackification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Quotients By Groupoids [R ⇒ V ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to Logarithmic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Temkin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Structure of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 63 64 64 64 65 66 67 67 67 68 69 70 70 72 73 74 74 75 75 76 76 76 76 78 78 78 79 81 81 81 82 82 84 85 87 87 88 89

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Precursors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Normal Crossings Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Toroidal Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Logarithmic Structures and Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Logarithmic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Logarithmic Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Logarithmic Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Morphisms of Logarithmic Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Logarithmic Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Logarithmic Blowings Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Logarithmic étaleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Stacks LogS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Constructions of LogS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Stacks Log and Logarithmic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90 90 91 93 93 95 98 104 107 107 109 112 114 115 116 119 122

Birational Geometry Using Weighted Blowing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dan Abramovich, Michael Temkin, and Jarosław Włodarczyk 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Place of Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 This Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Curves and Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 How to Resolve a Curve? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 How to Resolve a Surface? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Example: Whitney’s Umbrella . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 How to Resolve a Surface: Classical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Explaining the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Preview on Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Preview of Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Example: Blowing Up Whitney’s Umbrella x 2 = y 2 z. . . . . . . . . . . . . . . . . . . . . . 3.5 Definition of the Weighted Blowing Up Y  → Y . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Description of Y  → Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Determining the Center J Associated to an Ideal I in Examples . . . . . . . . . . 3.8 Describing the Blowing Up in the New Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Coefficient Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Defining JI in General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 What Is J ? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 123 124 124 124 125 125 125 126 126 126 127 128 128 129 130 130 131 132 132

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Elements of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Formal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Admissibility and Coefficient Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Key Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Birational Geometry and Blowing Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Birational Geometry Using Smooth Blowing Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Birational Geometry Using Weighted Blowing Up . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Strong Factorization of Toric Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 133 134 134 135 135 135 136 137

Relative and Logarithmic Resolution of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Temkin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Principalization and Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Classical Algorithm: A First Attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Classical Algorithm: The Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Resolution of Logarithmic Schemes: A First Attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Log Derivations and Log Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Log Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Lacuna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Logarithmic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Other Logarithmic Frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Resolution of Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Monomialization Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Dream Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Weighted Blowings Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Q-Ideals and Idealistic Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Rees Algebras and Rees Blowings Up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Non-logarithmic Weighted Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Logarithmic Weighted Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Resolution for Quasi-Excellent Schemes and Other Categories . . . . . . . . . . . . . . . . . . . 7.1 Reduction to Quasi-Excellent Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 140 142 144 144 146 150 151 153 154 160 162 166 168 168 174 178 178 179 183 188 189 190 195 196 199 204 207 208

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7.2 Extending the Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Desingularization for Quasi-Excellent Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Integral Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211 213 215 216

Weighted Resolution of Singularities. A Rees Algebra Approach . . . . . . . . . . . . . . . . . Jarosław Włodarczyk 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Applications of Weighted and Generalized Blow-Ups. . . . . . . . . . . . . . . . . . . . . . 1.2 Invariant and Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Motivating Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Cobordant and Stack-Theoretic Blow-Ups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Q-Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Stack-Theoretic Weighted Blow-Ups vs Cobordant Blow-Ups . . . . . . . . . . . . 3 Cobordant Blow-Ups of Rees Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Rational Rees Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 SNC Resolution and Principalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Main Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Coefficient Ideal for Rees Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Maximal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effective Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Properties of the Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Final Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Computations of the Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Examples of the Resolution in Positive Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Negative Examples in Positive Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Deformation to the Weighted Normal Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Characteristic-Free Resolution of Quasihomogeneous Singularities. . . . . . . 5.4 Characteristic-Free Resolution of Isolated Singularities . . . . . . . . . . . . . . . . . . . . 6 Generalized Blow-Ups and Cox Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Cox Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Relative Cox Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Cobordization of Morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Resolution of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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New Techniques in Resolution of Singularities: Open Problems . . . . . . . . . . . . . . . . . . Dan Abramovich, Anne Frühbis-Krüger, Michael Temkin, and Jarosław Włodarczyk 1 Computer Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Algorithmic Destackification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

219 221 223 225 227 230 230 241 243 243 247 247 254 264 275 289 290 292 298 298 301 304 305 307 307 308 310 313 315 319

319 320 321

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Theoretical Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Alternative Proofs of Our Resolution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Weighted Resolution in the Relative Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Extended Relative Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Resolution of Quasi-Excellent Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Weighted Resolution in Positive Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Other Blowup Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Log Weighted Resolution and Factorization of Birational Maps . . . . . . . . . . . 2.8 Strong Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

322 322 322 322 323 323 323 324 324 325

About the Authors

Dan Abramovich studied mathematics at Tel Aviv University and at Harvard, where he received his PhD in 1991. After a postdoctoral fellowship at MIT, he was on faculty at Boston University, where he was Sloan Research Fellow, before moving to Brown University where he is L. Herbert Ballou University Professor. He is Fellow of the American Mathematical Society and the American Association for the Advancement of Science. He works in birational Geometry, Moduli Spaces, and Arithmetic Geometry. His 2018 ICM talk at Rio de Janeiro focused on material described in the present book.

Anne Frühbis-Krüger works in computational algebraic geometry and singularity theory. She received her PhD from Kaiserslautern University in 2000. After a visiting professorship at FU Berlin and more than a decade as apl. Professor at Leibniz Universität Hannover, she moved to her current position at Universität Oldenburg in 2019. She has served as the spokesperson of the ‘Fachgruppe Computeralgebra’ for two periods and is actively involved in the development of the computer algebra systems Singular and OSCAR.

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About the Authors

Michael Temkin is the Maurice and Clara Weil Chair in Mathematics at the Hebrew University of Jerusalem. His main research interests lie within non-Archimedean geometry, birational geometry and the interplay between them. In particular, he is interested in resolution of singularities and semistable reduction problems. He graduated from the Weizmann Institute of Science in 2006, and after a postdoc at the University of Pennsylvania and a one-year membership at the IAS joined the Hebrew University of Jerusalem in 2010. Since 2020 he is serving as the editor in chief of the Israel Journal of Mathematics. Jarosław Włodarczyk works in birational geometry and resolution of singularities. He graduated from Warsaw University in 1993. After a visiting position in Ruhr University Bochum and a scholarship at Grenoble University, he joined Warsaw University where he worked till 2000. After that he moved to Purdue University, where he is now a Professor at the Department of Mathematics. Jarosław Włodarczyk was an invited speaker in ICM Madrid 2006, where he presented his work on the weak Factorization Theorem, playing an important role in Algebraic Geometry. He is also a recipient of numerous awards for his research contributions to birational geometry.

A Computational View on Hironaka’s Resolution of Singularities Anne Frühbis-Krüger

1

The Task of Resolving Singularities

To put the main topic, algorithmic aspects of resolution of singularities, into context and provide references to literature for the interested reader, we now very briefly discuss some facets of desingularization. In particular, we shall outline different variants of the task and then sketch the overall picture of an algorithmic approach to the problem. Notions used here and computational details will be discussed in significantly more depth in the subsequent sections.

1.1

Types of Resolution Tasks

Given a variety or scheme with singularities, the overall goal of resolution of singularities is to find a closely related non-singular variety or scheme and have full control over the passage from one to the other to easily relate their properties. We will make this very vague

This work has been supported by the German research foundation (DFG) through SFB/TRR 195. A. Frühbis-Krüger () Institut für Mathematik, Universität Oldenburg, Oldenburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_1

1

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A. Frühbis-Krüger

statement more precise in several variants, illustrating the differences between the variants along the way. Here, we roughly follow Kollàr’s more detailed textbook [15], section 3.1. Task 1.1 (Weak Resolution of Singularities) Let X be a variety. Find a non-singular variety .X and a proper birational morphism .X −→ X. As the following examples show, this still leaves a lot of possibilities and does not ensure very close relationship between X and .X : Example 1.2 In the first two examples, we specify the varieties as embedded varieties just for convenience of notation, but the targeted phenomena do not require the embedded setting, whereas the embedding is relevant in the last one. (a) Let .X = V (xw + yv + zu) ⊂ A6C which possesses a single singular point at the origin. Blowing up along the center corresponding to .x, y, z ⊂ C[x, y, z, u, v, w] provides a non-singular variety .X ⊂ A6C × P2C covered by three charts. However, .X differs from X not just at (the preimage of) the singular point, but at a subvariety of codimension 2 in X. (b) Let .X = V (x 2 z2 − y 4 + x 5 + z5 ) ⊂ A3C , of which the singular locus is the origin. Blowing up at the origin, we obtain .X ⊂ A3C ×P2C with ideal .x 2 z2 −y 4 +x 5 +z5 , xb− ya, xc − za, yc − bz. Passing to the chart .a = 0 of the projective space and then embedding into an .A3C as customary, the transform of X reads .V (c2 − y 4 + x + xc5 ), which is clearly non-singular. However, the exceptional divisor, the preimage in .X of the center, reads .V (x, c2 − y 4 ) = V (x, (c − y 2 )(c + y 2 )) in this situation and consists of two branches in the .(y, c)-plane touching tangentially. (c) Consider the previous example again, but take the perspective of blowing up the ambient space at the origin. Then the total transform .X of X is described by the same ideal as above, but passing to the same chart as before, we now have a total transform 4 2 4 5 .V (x (c − y + x + xc )) composed of the exceptional divisor .V (x) and the strict 2 4 transform .V (c − y + x + xc5 ) touching tangentially. Still not taking an embedding into account, one might want to exclude phenomena as in the first two examples of 1.2. For example (a) we can simply require that nothing outside the singular locus is changed. Formulating an appropriate condition to exclude (b), requires one more notion: simple normal crossing divisor. Definition 1.3 A divisor .D ⊂ W on a smooth variety W is called simple normal crossing, if each of its irreducible components is smooth and all intersections are transverse.

A Computational View on Hironaka’s Resolution of Singularities

3

This leads to the following more restrictive formulation: Task 1.4 (Strong Resolution of Singularities) Let X be a variety. Find a variety .X and a projective morphism .π : X −→ X such that (a) (b) (c) (d)

X is non-singular .π is birational −1 (X \ Sing(X)) −→ X \ Sing(X) is an isomorphism .π : π −1 .π (Sing(X)) is a simple normal crossing divisor. .

Now let some morphisms between varieties come into play: Example 1.5 ([15][3.3.4]) Consider the (non-smooth) morphism f : X1 −→ X2

.

(u, v) −  → (u2 , v 2 , uv) from .X1 = A2C to the quadric cone .X2 = V (xy − z2 ). Clearly, .X1 is already resolved, whereas a strong resolution .π : X2 −→ X2 is obtained by blowing up the origin. Unfortunately, there is no way to lift f to a morphism from .X1 to .X2 , because the image of the origin under f has been replaced by the exceptional curve under the blow-up. As we see from this example, we cannot hope for compatibility of the resolution with all morphisms. Restricting the considerations to smooth morphisms, however, we can clarify our task or wish further: Task 1.6 (Functorial Resolution of Singularities) For every variety X find a strong resolution of singularities .πx : X −→ X such that any smooth morphism .f : X −→ Y lifts to a smooth morphism .f  : X −→ Y  providing a cartesian square: f

X −→ Y 

.

↓  ↓ f

X −→ Y This task is called functorial resolution of singularities, because a solution to this task provides a functor from the category of varieties and smooth morphisms to the category of smooth varieties and smooth morphisms—the best we can hope for, given Example 1.5. Let us now come back to Example 1.2(c) and take additional data into account: Considering a variety X embedded in some smooth variety W , we may ask for a proper birational map .W  −→ W resolving singularities of X, i.e. for an embedded resolution

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A. Frühbis-Krüger

of singularities. This is the crucial setting appearing in any Hironaka-style resolution algorithm, as we will see toward the end of the section. Task 1.7 (Embedded Resolution of Singularities) Let W be a smooth variety and .X ⊂ W a variety. Find a smooth variety .W  and a proper birational morphism .π : W  −→ W such that 1. .π is an isomorphism outside the singular locus .Sing(X) of X 2. .π −1 (Sing(X)) is a simple normal crossing divisor (in W ). 3. .π induces a strong resolution of singularities of X. Principalization of an ideal sheaf .I on some smooth variety W is a problem of a similar flavour, as soon as we take into account the perspective of the subscheme of W defined by .I . Task 1.8 (Principalization of Ideals) Let .I ⊂ OX be a non-zero ideal sheaf on a smooth variety X. Find a smooth variety .X and a birational projective morphism .f : X −→ X such that .f ∗ I is a locally principal ideal sheaf on .X . Actually, it is more customary to ask for .f ∗ I to be the ideal sheaf of a normal crossing divisor on .X in the above task, but then the origin of the name is not so obvious. Remark 1.9 There are numerous other variants of resolution problems like e.g. resolving the locus of indeterminacy of a rational map. Also the question of desingularizing simultaneously a collection of schemes embedded in the same ambient space or desingularizing a family of varieties or schemes over some base space ‘in family’ arise in some contexts. Many approaches and algorithms covered here are also useful for such related problems. But the more additional properties should be preserved, the harder the problem becomes.

1.1.1 The Question of Existence The question of existence and construction of resolution of singularities has intrigued mathematicians for centuries and contributions are so numerous that we only mention very few, selected due to particular relevance for the later algorithmic considerations. We will take an algorithmic approach along the lines of Hironaka’s groundbreaking article [13], which completely solved the task of (embedded) resolution of singularities in characteristic zero and which was made constructive by Villamayor [21] and independently by Bierstone and Milman [1]. In positive and mixed characteristic, the problem is still wide open— except in small dimensions or except for particular classes like binomial ideals. We refer the interested reader to textbooks on resolution of singularities like [15] and [6] and to the proceedings volume [12] for more aspects and approaches. For the toric case, textbooks on toric geometry cover desingularization in the flavor of fan subdivision from a different point of view also not covered in this article.

A Computational View on Hironaka’s Resolution of Singularities

5

1.1.2 Curves The simplest setting for desingularization tasks is the case of curves with singularities. Here two algorithmic possibilities immediately come to mind: normalization and blowing up. In the first case, having no singularities in codimension 1 already implies being non-singular for curves, whereas numerous normal varieties in higher dimension possess singularities. Additionally normalization only solves the tasks of weak and strong resolution of singularities, but not of embedded desingularization, since it does not take the embedding into account. The use of a sequence of blow-ups on the other hand is very simple for curves: the singular locus is a finite set of points and we can blow up at these points. However, even in extremely simple cases like e.g. an .A3 plane curve singularity one blow-up at the only singular point does not suffice to resolve the singularity, it only makes it ‘simpler’: Example 1.10 Consider the .A3 singularity defined by .y 2 + x 4 − x 5  ⊂ OA2 . It possesses C a single isolated singularity at the origin, which locally looks like two touching parabolas, see the green curve in Fig. 1. Blowing up at the only singular point and passing to the two affine charts of .P2 as usual, we see a single singular point at the origin of one chart with local ideal .y12 + x 2 − x 3  (again green, two branches meeting transversally) and exceptional divisor .V (x) (red) in Fig. 2. The other chart does not contain any singularity (the red and green curves do not touch in the picture). For the second blow up, we first notice that the chart without singularity will not change and a local consideration in the other affine chart suffices. So blowing up the origin of this chart, we obtain again two charts, in which we now see the strict transform of the original curve under the sequence of blow ups (green), the strict transform of the first exceptional Fig. 1 An .A3 plane curve singularity

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A. Frühbis-Krüger

Fig. 2 The .A3 plane curve singularity after the first blow up

Fig. 3 The .A3 plane curve singularity, charts of the second blow up

divisor (red) and the new exceptional divisor (purple) in Fig. 3. The following shows this example in a session in the computer algebra system Oscar [20].1 julia> W = affine_space(QQ,2) # affine 2-space W Spec of Multivariate Polynomial Ring in x1, x2 over Rational Field julia> (x,y) = gens(OO(W)) 2-element Vector{QQMPolyRingElem}: x1 x2

# naming of variables

julia> f = y^2 + x^4 -x^5 -x1^5 + x1^4 + x2^2

# polynomial defining X

julia> X = subscheme(W,f) # our singular variety X Spec of Quotient of Multivariate Polynomial Ring in x1, x2 over Rational...

1 Note that we are using methods from the experimental folder of the development version (0.12,

March 2023). So output formatting might change slightly before the features become official.

A Computational View on Hironaka’s Resolution of Singularities

7

julia> ## determine its singular locus -- ’;’ suppresses output Sing, inc_sl = singular_locus(X); julia> Ising = image_ideal(inc_sl) ideal(x2, x1)

# ideal of singular locus

julia> # blow up at the only singular point blmap = blow_up(X,Ising) # blow up at ideal in OO(X) Blow up of a Covered Scheme with 1 Charts leading to a Covered Scheme ...

julia> ## now look at the data of the blow-up codomain(blmap) # X as a covered scheme covered scheme with 1 affine patches in its default covering julia> X1 = domain(blmap) # transform of X covered scheme with 2 affine patches in its default covering julia> E = exceptional_divisor(blmap) # exceptional divisor, short Effective Cartier Divisor on Covered Scheme with 2 Charts

julia> show_details(E) # exceptional divisor, long Effective Cartier Divisor on Covered Scheme with 2 Charts: Chart 1: ideal in Localization of Quotient of Multivariate Polynomial Ring in (s1//s0), x1, x2 over Rational Field by ideal(-x1^5 + x1^4 + x2^2, -(s1//s0)*x2 + x1, (s1//s0)*x1^4 - (s1//s0)*x1^3 - x2, (s1//s0)^2*x1^3 - (s1//s0)^2*x1^2 - 1, (s1//s0)^3*x1^2*x2 - (s1//s0)^3*x1*x2 - 1, (s1//s0)^4*x1*x2^2 - (s1//s0)^4*x2^2 - 1, (s1//s0)^5*x2^3 (s1//s0)^4*x2^2 - 1) at the multiplicative set powers of QQMPolyRingElem[1] generated by [x2] Chart 2: ideal in Localization of Quotient of Multivariate Polynomial Ring in (s0//s1), x1, x2 over Rational Field by ideal(-x1^5 + x1^4 + x2^2, (s0//s1)*x1 - x2, -(s0//s1)*x2 + x1^4 - x1^3, -(s0//s1)^2 + x1^3 - x1^2, -(s0//s1)^3 + x1^2*x2 - x1*x2, -(s0//s1)^4 + x1*x2^2 - x2^2, -(s0//s1)^5 - (s0//s1)*x2^2 + x2^3) at the multiplicative set powers of QQMPolyRingElem[1] generated by [x1]

julia> Sing2,inc_sl2 = singular_locus(X1);

# singular locus after blow up

julia> Ising2 = image_ideal(inc_sl2) Ideal Sheaf on Covered Scheme with 2 Charts

# ideal sheaf of it

julia> blmap2 = blow_up(Ising2) # blow up at ideal sheaf Blow up of a Covered Scheme with 2 Charts leading to a Covered Scheme with 3 Charts

julia> X2 = domain(blmap2) # transform covered scheme with 3 affine patches in its default covering julia> Sing3,inc_sl3 = singular_locus(X2);

# singular locus

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julia> is_empty(Sing3) true

# is emtpy

Note that for the first blow up all input data is still in a single polynomial ring, whereas the first blow-up map is already a map of covered schemes and for this purpose the original affine scheme was automatically encoded in a covered scheme with trivial covering. From this point on everything is encoded in terms of covered schemes.

1.1.3 The Challenge of Spotting Improvement This simple example has been chosen to illustrate the improvement in the strict transform with the bare eye. In general, we need to algorithmically measure improvement and decide, when we are finished—determined by data at each of the points of the curve. Luckily, we know that a plane curve is non-singular, if it can be described locally at each of its points by a power series of order 1. Therefore the multiplicity of the curve at a point is a natural first guess for such a measure. But it does not suffice, as we directly see in first chart of the first blow-up in Example 1.10, where the situation (obviously) improved even though the multiplicity at the only singular point is still 2 as it was in the only singular point at the beginning. The philosophy behind spotting the improvement in this example is seeing y as the main variable and considering the orders of the coefficients of powers of y (here only of .y 0 ) in the defining power series in a suitable way. This choice of a main variable is a toy instance of a central ingredient to choosing centers of blow-ups and will be made precise for the general setting in characteristic zero in Sect. 5. 1.1.4 More Trouble in Higher Dimension Moving one dimension higher, the situation already becomes much more complicated: the singular locus of a surface can have dimension one and may exhibit all kinds of bad properties we can imagine for curves. Historically, there have again been different approaches to deal with this, like using normalization to achieve a zero-dimensional singular locus [17] or being guided by resolution of the branch locus of a projection to the plane [14] or first resolving the singular points of the singular locus followed by blowing up at the (then smooth) singular locus of the intermediate result. The philosophy of the last mentioned approach is the underlying idea of Hironaka-style resolution of singularities: an induction on the dimension guiding the choice of the upcoming center.

1.2

The Philosophy of Hironaka-Style Resolution

Before going into algorithmic details in the subsequent sections, we shall now discuss the overall structure of Hironaka-style resolution of singularities, emphasizing how to break this down to manageable subtasks which will later be discussed from the algorithmic point of view one at a time. Let us start by stating Hironaka’s result in a variant which already allows a glimpse at the construction:

A Computational View on Hironaka’s Resolution of Singularities

9

Theorem 1.11 ([3, 13]) Given a quasi-projective variety X over a field of characteristic zero, there is a finite sequence of blow-ups πr−1

π1

π0

π : Xr −→ · · · −→ X1 −→ X0 = X

.

at suitably chosen smooth centers .Ci such that • .Xr is smooth • .π is an isomorphism outside the singular locus of X • the exceptional divisor of .π has simple normal crossing This shows a rather straightforward overall structure of the algorithm as illustrated in Fig. 4 But it also illustrates two distinct algorithmic tasks: the blow-up itself, which is rather straight forward and will be covered in Sect. 3, and the choice of center, which is the real heart of the algorithm and occupies most of the other parts of the chapter. The guiding idea in the choice of center is to blow up at the ‘worst’ points first, but this is subtle as the following examples show: Example 1.12 Let us consider the Whitney umbrella .V (x 2 − y 2 z) ⊂ A3C . Its Jacobian ideal is .x, yz, y 2  = x, y ∩ x, y 2 , z, implying that the singular locus is the z-axis. But one might argue that the worst point is the origin in view of the embedded component of the ideal. Fig. 4 Flow chart of the overall structure of a Hironaka-style resolution of singularities

variety X

resolved?

no

Determine Center

Blow-Up

yes

return

10

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Blowing up at the origin and then passing to the chart with exceptional divisor .V (z), we see a strict transform of the form .V (x12 − y12 z) which is again a Whitney umbrella. So we did not achieve any improvement. In the chart with excpetional divisor .V (y), we see an .A1 singularity at the origin, whereas no singularity is visible in the last chart. We might even argue that we made the situation worse as the singular locus now has a second component. Blowing up at the z-axis, resolves the singularity directly. So the larger center was the “right” choice in this setting. Example 1.13 If we consider a slightly modified version of the previous example, the situation has changed: .V (x 2 − y 2 z2 ) ⊂ A3C possesses a singular locus consisting of the y- and the z-axis, i.e. two crossing lines meeting at the origin. As we want to avoid singular centers, we cannot blow up at the whole singular locus. Choosing one of the lines at random certainly provides a good center, but the random choice by itself raises new problems: We could be somewhere in the middle of a sequence of blowing ups and see the same singularity in several charts. Then random choices of the center easily lead to inconsistencies between the charts and render glueing impossible. So we end up with only one choice left: the origin, which is the singular locus of the singular locus. This blow up provides us with three charts, one of which does not see a singularity, the other two basically show the same setting with roles of y and z reversed: The strict transform possesses the same singularity .V (x12 − y 2 z12 ) as the original variety, but now one of the two lines in the singular locus lies inside the exceptional divisor, the other one does not. In other words, the blow up has separated the two original lines, which can now be blown up independently of each other and in the local situation we have sufficient information to avoid a random choice. As already hinted in the preceding section, a good first ingredient to measuring, how ‘bad’ the situation at a given point is, can be some generalization of the order of a power series. To achieve such a generalization, Hironaka introduced standard bases,2 a particular choice of a system of generators for an ideal in a local ring (power series ring). In a nutshell, the powers of the maximal ideal .m of a local ring R induce a filtration on R and the initial part of an element .f ∈ R will be the class of f in .ms /ms+1 with the smallest s to allow a non-zero image. A standard basis of an ideal .I ∈ R is a particular system of generators with the main property that the set of initial forms of these generators generates the same ideal as the set of initial forms of all elements of I in the associated graded ring  i i+1 of R. The degrees of the initial forms, in increasing order, can .gr m (R) = i≥0 m /m 2 The term standard basis is nowadays understood as the local analogue to the well-known term

Groebner basis, which will be briefly discussed as algorithmic background in Sect. 2. For a computer algebra textbook with this point of view, we recommend [9]. If we need to stress the subtle difference, which lies in considering the initial forms (Hironaka) or the initial terms (computer algebra), we shall state this explicitly.

A Computational View on Hironaka’s Resolution of Singularities

11

be compared lexicographically, alternatively similar information can be collected from the Hilbert-Samuel function of the local ring .R/I . Either way, the computation at a given point is not difficult, but may take some time and space. When asking for the points, where the maximal value is attained, the problem becomes much more involved and is in general hard to compute. So for practical reasons, the preference can be put on a coarser first ingredient, the lowest degree of an initial form appearing in an ideal. This is the order of the ideal at a point and will be discussed in detail in Sect. 4. Revisiting one of the previously considered examples immediately shows that this ingredient is a good starting point, but by far not enough to guarantee smooth centers: Example 1.14 (Example 1.13 Revisited) The polynomial .x 2 − y 2 z2 ⊂ C[x, y, z] attains its maximal order 2 at any point of .V (x, yz). This is just the singular locus, which we had computed before, and consists of two crossing lines, which has a singularity at .V (x, y, z). But how do we extract additional information from the polynomial to further narrow down this locus? One possible way to proceed is the following: Concentrate on the smooth hypersurface 3 .Y = V (x) ⊂ A , which contains the locus of maximal order of X, and set principalization of some suitable new ideal as an intermediate goal, in our case .Inew = y 2 z2  ⊂ OA3 /x. Choose centers, which principalize the ideal and resolve .V (Inew ) ⊂ V (x) ⊂ A3C , and use them to achieve a smooth locus of maximal order of X. The approach of the example is indeed the main idea of the descending induction on the dimension: Locally the locus of maximal order is contained in a suitable smooth hypersurface, a so-called hypersurface of maximal contact, which is guaranteed to exist in characteristic zero. The creation of the new ideal or new variety on this hypersurface is in general much more complicated than in the easy example and involves viewing a generator of the ideal of the hypersurface as main variable and collecting coefficients w.r.t. this variable. This key idea is explored in Sect. 5. So at present, we have the following recursive structure for the choice of center described in Fig. 5 below. But this view is still oversimplified: hypersurfaces of maximal contact exist only locally, so the step of finding hypersurfaces of maximal contact involves a refinement of the covering by affine charts—a crucial step in the computation. Another detail which we have not yet seen, except in Example 1.13, is the use of certain exceptional divisors to encode the history and disambiguate certain choices. To this end, selected exceptional divisors need to be taken into account at each level of the recursion.3 We will discuss this at the end of the article in Sect. 6.

3 This is a detail whose relevance does not pertain to the weighted resolution in the other parts of

this volume, whence it is only covered very briefly here.

12

A. Frühbis-Krüger

Fig. 5 Flow chart describing the general structure of the choice of center

ideal IX

Determine locus of maximal order

is regular?

yes

return

no Find hypersurface of maximal contact

Determine Inew

A less simplified structure of the algorithm is shown in Fig. 6. Remark 1.15 The algorithmic Hironaka-style approach outlined here breaks down in positive characteristic, as the existence of hypersurfaces of maximal contact does not hold in positive characteristic and the order of the coefficient ideal is not semicontinuous under blow-ups in positive characteristic. However, having a characteristic zero implementation at hand, and wishing to compute an example in positive characteristic, it is always worth a try, since the problems in positive characteristic are not ubiquitous, but only occur in specific settings. In general, a convenient way to phrase Hironaka-style desingularization is to formulate it by means of an invariant which serves as the measure, how bad a point is, and to choose the locus of maximal value of this invariant as a the upcoming center. We have already seen the basic ideas behind this invariant and its recursive structure. What should be added to conclude this sections is a summary on the properties of the invariant. Remark 1.16 Let W be a smooth scheme and .X ⊂ W a subscheme for which the task of embedded resolution of singularities is to be solved. Then a resolution invariant, controlling the desingularization process and choosing the upcoming center as the locus of its maximal value, needs to satisfy the following properties:

A Computational View on Hironaka’s Resolution of Singularities Fig. 6 Flow chart describing the general structure of the choice of center

13

ideal IX

Determine locus of maximal order

is regular?

no

yes

Suitably recombine local results

return

Locally find hypersurfaces of maximal contact

Locally determine Inew

(a) Zariski upper semicontinuity This condition ensures that the locus of maximal value is a Zariski-closed set. i.e. it is a subscheme at which we can blow-up. (b) infinitessimal upper semicontinuity (under blow-ups at maximal locus) If the invariant is to measure the progress under a sequence of blow-ups, it should not increase under the blow-ups it prescribes. (c) non-singularity of maximal locus Surely, we do not want to introduce singularities, but remove them. (d) normal crossing of maximal locus with exceptional divisors If the center is not normal crossing with the exceptional divisors, we cannot expect the newly arising divisor to be normal crossing with them after the blow up. (e) decrease of maximal value under blow-up along maximal locus For termination of the algorithm, not only the decrease of the maximal value is crucial, but also exclusion of infinite sequences of invariant values converging to the minimal value without attaining it after finitely many steps. (f) minimal value attained at already resolved points This is the termination condition for the algorithm.

14

A. Frühbis-Krüger

Remark 1.17 To a certain extent, the weighted resolution algorithm covered in later chapters of this book, also follows this philosophy. But there are important differences: instead of usual blow ups weighted blow ups are used, exceptional divisors do not come into play for the choice of center and the result is in general a stack. The implications of the last difference are discussed in detail in Remark 6.7 in the chapter on stacks.

2

Computational Basics on Groebner Bases and Ideal Operations

Before discussing the computational tasks of algorithmic resolution of singularities in the subsequent sections, we lay some foundations. Let us first recall basic tools from computational commutative algebra: monomial orderings, Groebner bases, elimination and applications thereof. In particular intersections and quotients of ideals will later be the building blocks for more sophisticated computations. Throughout this section, we assume that we are working in a polynomial ring .R = K[x1 , . . . , xn ] =: K[x] over a (computable) field K. After defining a suitable wellordering on the set of monomials .Mon{x} := {x α ∈ R | α ∈ Nn } of R, Groebner bases will be introduced as a black box, i.e. without discussing their computation. For the scope of this article, we only need to know that there are algorithms by which they can be computed. The interested reader is referred to standard textbooks like for instance [5, 9]. Here, we strictly focus on those tools arising from applications of Groebner bases, which will be needed later on to make the resolution algorithm computable.

2.1

Groebner Bases in a Nutshell

Already the wish to write down or represent polynomials in some fixed way, imposes the need to choose an ordering on the set of monomials of R. But for the computational purposes of Gröbner Bases techniques, we additionally need to make sure that the chosen total ordering is compatible with multiplication by a monomial. To simplify the discussion, we will assume that the ordering is a well-ordering, that is .1 = x 0 is the smallest monomial. Remark 2.1 Note, however, that this last condition is omitted for computations in localizations as is discussed in detail in [9], Chapter 1, as far as localizations at the origin are concerned. Localizations at K-points (where K denotes the underlying field) can be performed by translation of the point to the origin. Localizations at maximal ideals not corresponding to a K-point are algorithmically significantly harder, see [18]. In the case of orderings which are not a well-ordering, the term standard basis is used, although there are subtle differences to the notion of standard basis introduced by Hironaka.

A Computational View on Hironaka’s Resolution of Singularities

15

For our context in this section, a suitable notion of ordering on the set of monomials of R is the following: Definition 2.2 A monomial ordering .> on .Mon(x) is a total ordering on the set of monomials in R which is a well-ordering and satisfies the additional property ∀x α > x β ∈ Mon(x), ∀x γ ∈ Mon(x) : x α · x γ > x β · x γ

.

Example 2.3 To see the effects of different monomial orderings, we consider three of the most commonly used ones for the example of the set of monomials .{x23 , x1 x2 , x12 }: Lexicographical

The ordering .>lex is defined by

x α >lex x β :⇐⇒ ∃1 ≤ k ≤ n : (αi = βi ∀i < k) AND (αk > βk )

.

The comparison of the monomials from above provides: x12 >lex x1 x2 >lex x23 .

.

Degree-lexicographical

The ordering .>Dp is defined by

x α >Dp x β :⇐⇒ (|α| > |β|) OR ((|α| == |β|) AND (x α >lex x β ))

.

The comparison of the monomials from above provides: x23 >dp x12 >dp x1 x2

.

Weighted-lexicographical defined by

The ordering .>wp with respect to weights .(a1 , . . . , an ) is

x α >wp x β :⇐⇒ (wdeg(α) > wdeg(β)) OR

.

((wdeg(α) == wdeg(β)) AND (x α >lex x β ))  where .wdeg(α) := ni=1 ai αi . The comparison of the monomials from above provides for .>wp with weights .(3, 4): x23 >wp x1 x2 >wp x12

.

where the weighted degrees in our example are .wdeg(x23 ) = 12, .wdeg(x1 x2 ) = 7 and 2 .wdeg(x ) = 6. 1

16

A. Frühbis-Krüger

Given a monomial ordering, it makes sense to talk about the largest monomial in a given polynomial and even about the set of all monomials appearing as largest monomial of a polynomial in a given ideal: Notation 2.4 Let .> be a monomial ordering, .f ∈ R a polynomial and .G ⊆ R any subset. Then we will use the following notation: LT (f ) := largest monomial in f w.r.t. > leading term L(G) := LT (g) | g ∈ G leading ideal

.

Remark 2.5 Due to the compatibility of a monomial ordering with multiplication with monomials, we know that .LT (f g) = LT (f ) · LT (g) Example 2.6 Let .R = Q[x, y], let us choose degree-lexicographical ordering and let .f = x 3 + y 2 . We now consider .G = {f } and .I = f . We immediately see that .LT (f ) = x 3 and hence .L(G) = x 3 . On the other hand, any element .h ∈ I is of the form .h = f · g for some .g ∈ R, whence .x 3 | LT (h) and thus .L(I ) = x 3 . So in this case, .L(G) = L(I ) for the set of generators G. But this is obviously not always the case: Consider .G = {x 3 −y 2 , x 3 −z} ⊆ Q[x, y, z], .I = G ⊆ Q[x, y, z] and again degreelexicographical ordering. In this case both generators have the same leading term .x 3 , so .L(G) = x 3 , but clearly .y 2 − z ∈ I and hence .y 2 ∈ L(I ). Thus .L(G) = L(I ). The property .L(G) = L(I ) in the preceding example clearly does not depend on the ideal, but on the chosen set of generators. It is the property of G being a Groebner basis for the ideal I : Definition 2.7 Let .> be a monomial ordering on .Mon{x} and let .I ⊆ R be an ideal. Then a Groebner basis of I w.r.t. the ordering .> is a finite set .G ⊂ Mon{x} such that L(G) = L(I ).

.

Several ideal-theoretic properties can be read off combinatorially from the leading ideal of a given ideal and hence be computed from the elements of a Groebner basis: most prominently the dimension of the ring .R/I and the Hilbert polynomial.4 This reduction to combinatorial methods is computationally very handy, but will not be our

4 Caution: When referring to documentation of computer algebra systems, you also encounter

‘dimension of I ’ in the sense of dimension of .R/I .

A Computational View on Hironaka’s Resolution of Singularities

17

main focus here. Instead, the necessary applications for our purpose are deciding ideal membership and radical membership, computing elimination of variables, intersection of ideals and saturation. The applications which we are aiming at, are computing blow ups and determining centers in algorithmic resolution of singularities. A key fact on the route to these applications is the fact that given a ring .R/I , a Groebner basis .G = {g1 , . . . , gs } of I and a polynomial .f ∈ R, we can algorithmically find a representative r of the class of f in .R/I by Buchberger’s normal form algorithm (see e.g. [5, 9] for a discussion of the algorithm) such that: • either .r = 0 or .LT (r) ∈ L(G)  • .∃a1 , . . . , as ∈ R : f − r = si=1 ai gi and for each i either .ai = 0 or .LT (ai gi ) ≤ LT (f ). In this case, r is called a normal form of f w.r.t. I . This can be understood as a first step in the direction of division with remainder. Despite still being far from uniqueness, normal forms already allow testing for ideal membership:

2.1.1 Task: Ideal Membership Test Knowing that a normal form can be computed, it is easy to check whether a given f is an element of I or, in other words, whether f is in the class of 0 in .R/I : Compute a normal form r of f w.r.t. a standard basis G of I . Then .f ∈ I iff .r = 0. 2.1.2 Task: Unique Representative w.r.t. I Given a normal form r of a polynomial f w.r.t. a Groebner basis G and imposing the stricter condition that not only the leading term, but any term of r is not contained in .L(G), we call r a reduced normal form. A reduced normal form can be computed algorithmically again by Buchberger’s normal form algorithm and additionally using socalled tail reduction (cf. [9], chapter 1 for a treatment of this feature). This is possible, because the underlying ordering is a well-ordering. Reduced normal forms are known to be unique ([9][1.6.7]), whence the computation of a reduced normal form can be interpreted as a generalization of division with remainder to the case of Groebner bases. 2.1.3 Task: Radical Membership Test √ By Rabinovitch’s trick we know that .f ∈ I ⊆ K[x] iff .1 ∈ I  + 1 − tf  ⊆ K[x, t]. Therefore radical membership can be decided by the corresponding ideal membership test in the polynomial ring with one additional variable.5

5 In practice, this is significantly cheaper than computing the radical of the original ideal and then

testing for ideal membership.

18

2.2

A. Frühbis-Krüger

Applications: From Elimination to Saturation

Arguably the most important strength of Groebner Basis techniques is that they make numerous standard operations on ideals (and modules) computable. The key to many of these is intersection with subrings, the first application which we will consider here:

2.2.1 Task: Elimination of Variables Let .I ⊆ K[x]. The task of computing .I ∩ K[xr+1 , . . . xn ] is usually referred to as the elimination of the variables .x1 , . . . , xr and can geometrically be interpreted as the closure of the projection of .V (I ) to .An−r K . Its computation is based on a suitable choice of monomial ordering and a Groebner basis computation: Definition 2.8 A monomial ordering on .Mon(x) is called an elimination ordering for x1 , . . . , xr , if the following property holds:

.

∀f ∈ K[x] : LT (f ) ∈ K[xr+1 , . . . , xn ] ⇒ f ∈ K[xr+1 , . . . , xn ].

.

Of course, elimination orderings exist: For instance, lexicographical ordering satisfies the above condition for each .1 ≤ r ≤ n. Unfortunately, this ordering is one of the most expensive ones in terms of time and space consumption in practice and thus also a good reminder that we cannot expect elimination orderings to be cheap. Lemma 2.9 Let .> be an elimination ordering for .x1 , . . . xr on .Mon(x), let .I ∈ K[x] and let G be a Groebner basis of I w.r.t. .>. Then G := {g ∈ G | LT (g) ∈ K[xr+1 , . . . , xm ]}

.

is a standard basis of .I ∩ K[xr+1 , . . . , xm ] w.r.t. the ordering induced on the smaller set Mon(xr+1 , . . . , xn ) by .>.

.

Of course, .G is a set of generators for .I ∩ K[xr+1 , . . . , xm ] and hence the task is solved. This is indeed an extremely versatile tool enabling us to perform important ideal operations as we see in the following examples.

2.2.2 Task: Intersection of Ideals Given two ideals .I1 = f1 , . . . , fr  and .I2 = g1 , . . . , gs  in .K[x], the computation of a set of generators for the ideal .I1 ∩ I2 can be reduced to an elimination of variables for a suitably chosen ideal:

A Computational View on Hironaka’s Resolution of Singularities

19

Lemma 2.10 Given .I1 , I2 as above, set J := tf1 , . . . , tfr , (1 − t)g1 , . . . , (1 − t)gs  ⊆ K[t, x].

.

Then .I1 ∩ I2 = J ∩ K[x].

2.2.3 Task: Quotient of Ideals Given two ideals .I1 , I2 ⊂ K[x], we can reduce the computation of the ideal quotient .I1 : I2 to the computation of intersections of suitably chosen ideals: Lemma 2.11 Let .I1 ⊂ K[x], .g ∈ K[x] and .G = gh1 , . . . , ghm  a Groebner basis of I1 ∩ g. Then

.

I1 : g = h1 , . . . , hm .

.

Let .I2 = g1 , . . . , gs , then I1 : I2 =

s 

.

(I1 : gi ).

i=1

2.2.4 Task: Saturation w.r.t. an Ideal Let .I1 , I2 ⊂ K[x]. Iteratively applying the ideal quotient .(. . . (I1 : I2 ) · · · : I2 ) leads to an ascending chain of ideals I1 ⊆ I1 : I2 ⊆ I1 : I22 ⊆ · · · ⊆ I1 : I2k ⊆ . . .

.

which becomes stationary at some point, i.e. there is a .k0 ∈ N such that .I1 : I2k0 = I1 : I2k0 +N for all .N ∈ N, because the polynomial ring .K[x] is noetherian. In general .I1 : I2k0 is called the saturation of .I1 by .I2 and is denoted by .I1 : I2∞ . Ideal quotients and saturation will be key techniques for computing weak, controlled and strict transforms under blow-ups, a task which we address Sect. 3.

3

Blowing Up: From Definition to Computation

As already outlined in Sect. 1 the main workhorse of Hironaka-style resolution of singularities is blowing up. After briefly recalling basic facts to fix notation, we will concentrate on methods to compute it and discuss efficiency. Very important in the context of desingularization are notions of transform, like the well-known total and strict transform, but also some intermediate objects like the weak or the controlled transform. These concepts are readily demystified by explicit computation.

20

3.1

A. Frühbis-Krüger

Blowing Up at an Ideal Sheaf I on a Scheme W

The varieties and schemes which we are going to blow up during the algorithm need to be implemented with the help of affine algebras. It is thus natural to think about them in terms of a given affine covering. So we will quickly reduce our considerations to the affine setting. From standard textbooks on Algebraic Geometry (for instance [10], pp. 163–168), we know the definition and properties of blowing up: Definition 3.1 Let W be a scheme and .I a coherent ideal sheaf on W . Then the blow-up of W at .I is the scheme W˜ = P rojW (



.

I d ),

d≥0

where .I 0 is set to .OW for simplicity of notation. The induced morphism πI : W˜ −→ W

.

is usually referred to a the blow-up morphism of W in .I or for short as the blow-up of W in .I. Denoting the closed subscheme of W defined by .I by Y , the scheme .W˜ is often also referred to as the blow-up of W at Y . Recall 3.2 Let .I be a coherent sheaf of ideals on a noetherian scheme W defining a closed subscheme .Y ⊂ W and let .π : W˜ −→ W be the blow-up of W at .I. 1. (Exceptional divisor) −1 I · O , the inverse image sheaf of .I under .π , is an invertible sheaf on .W ˜ . Its zero .π W˜ locus is usually referred to as the exceptional divisor of the blow-up. π 2. The morphism .π is an isomorphism outside of Y , i.e. .π −1 (W \ Y ) ∼ = W \ Y. 3. (Universal property of blow-up) Let .f : Z −→ W be any morphism satisfying that .f −1 I · OZ is an invertible sheaf on Z. Then there exists a unique morphism .g : Z −→ W˜ factoring f : Z

∃!g

f

W˜ π

W

A Computational View on Hironaka’s Resolution of Singularities

21

4. (strict transform) If .Z ⊂ W is a closed subscheme and .πz denotes the blow-up of Z at .IOZ , the universal property provides: Z˜ → W˜

.

πz ↓

↓π

Z → W where the morphism .Z˜ → W˜ is again a closed immersion. .Z˜ is referred to as the strict transform of Z under .π . In contrast to this .π ∗ (Z) is called the total transform of Z under .π .

3.1.1 Blowing Up an Affine Chart Passing to affine charts allows us to translate this into algorithmic steps: Let .U ⊂ W be an affine open subset, denote the affine algebra .(U, OW ) of global sections by A and let J be the ideal .(U, J ) = f1 , . . . , fm  ⊆ A. Then the blow-up of U at J is just π −1 (U ) = P roj (



.

J d ).

d≥0

We would, however, prefer a description of this ring as a quotient of some polynomial ring by an appropriate ideal, as this is convenient for computational purposes. To achieve this, we perform the following elimination: Consider the A-algebra homomorphism  : A[y1 , . . . , ym ] −→



.

J n t n (⊆ A[t])

n≥0

defined by .(yi ) = tfi . Remark 3.3 (Algorithmic Side Remark) The additional variable t in the target of . is used to keep track of the grading, because otherwise ambiguities arise when trying to decide e.g. whether the element .f · f should be seen as an element of J or of .J 2 for a given .f ∈ J . From the definition of . it is clear that .⊕n≥0 J n is isomorphic to the ring .A[y1 , . . . , ym ]/Ker() and we can hence describe the geometric situation by means of the embedding π −1 (U ) ∼ = V (Ker()) ⊆ Spec(A) × Pm−1 .

.

The projective space over .Spec(A) will again be covered by the standard affine charts D(yi ) giving rise to an affine covering of .π −1 (U ).

.

22

A. Frühbis-Krüger

Given this presentation of .W˜ in terms of quotients of polynomial rings, it is not difficult to write down the ideal of the excpetional divisor explicitly: .IE = f1 , . . . , fm A[y]/Ker() . Example 3.4 We illustrate the computation of blowing up by means of the example of an E6 surface singularity .Z = V (x 2 + y 3 − z5 ) ⊆ A3C and the center .V (x, y, z), which is the only singular point of this isolated singularity. In view of the above construction, we have

.

A = C[x, y, z]/x 2 + y 3 − z5 

.

J = x, y, z Thus the .C-algebra morphism . reads .

  C[x, y, z]/x 2 + y 3 − z5  [a, b, c] −→ (C[x, y, z]/x 2 + y 3 − z5 )[t] a −→ tx b−  → ty c−  → tz

and an easy direct computation leads to   Ker() = ay − bx, az − cxbz − cy ⊂ C[x, y, z]/x 2 + y 3 − z5  [a, b, c].

.

Passing to the usual charts of projective space, we see Chart a = 0 . b = 0 c = 0

IZ˜ locally 1 + ( ab )3 x − ( ac )5 x 3  ⊂ C[x, ab , ac ] ( ab )2 + y + ( bc )5 y 3  ⊂ C[y, ab , bc ] ( ac )2 + ( bc )3 z + z3  ⊂ C[z, ac , bc ]

exc. divisor locally V (x) V (y) V (z)

For simplicity of notation and for keeping track through sequences of blow ups, one often renames the variables, e.g. .y1 = ab and .z1 = ac in the first chart. We will do this at several places in this section without mentioning it again.

3.1.2 Efficiency Considerations I: Number of Variables In each of the charts of the preceding example, we saw that some variables could be eliminated immediately, allowing a description of the transform .π −1 (U ) in the respective chart by means of a quotient of a polynomial ring in fewer variables. This is not only desirable in computations by hand, where it simply helps to keep the notes more compact, but also on the computer, where it can significantly affect the cost of calculations. Algorithmically selecting the variables for this very particular elimination of variables,

A Computational View on Hironaka’s Resolution of Singularities

23

which needs to still provide an isomorphism, requires a heuristic considering the linear part of the generators, of which the only difficulty is to keep the computational costs low. Remark 3.5 (Computational Point of View) Passing to affine charts leads to a significantly higher number of computational tasks, but reduces the number of variables involved in each of these as we saw e.g. in Example 3.4. Keeping in mind the doubly exponential complexity of Groebner bases w.r.t. the number of variables, the practical observation that each of the computations in charts is (in general) much easier than staying in .A[y1 , . . . , ym ] is not surprising. But overall it is a tradeoff between time and space, which breaks up large computations into several smaller ones which can be handled sequentially or maybe in parallel, if adequate hardware and infrastructure are available.

3.1.3 Working with Affine Coverings Work with covered schemes has always been very tedious in computer algebra systems, because traditionally they only provided computations in polynomial rings and the user had no tools to keep track of his charts and their glueing. Recently some of the systems have made an effort to include functionality in this direction; for instance, Oscar contains an implementation of basic functionality for affine, projective and covered schemes in its newest version including a blow-up functionality. This allows to handle a covered scheme as a single object, but also access each chart directly. The examples in Sect. 3.4 also include some sample Oscar sessions. 3.1.4 Efficiency Consideration II: Centers There is a further important setting, in which choosing a variant with a higher number of charts leads to an easier setting in each of the charts: the case of reducible smooth centers. The following example illustrates this setting with a center consisting of three reduced points, which can, of course, be blown up all at the same time or one after the other in any order: Example 3.6 We choose the center .{(0, 0), (1, 0), (0, 1)) ⊂ A2C , which corresponds to the ideal xy, x 2 − x, y 2 − y ⊂ C[x, y].

.

Notice that we cannot describe this ideal of three points in two-dimensional space with fewer than 3 generators and hence need to add new variables .a, b, c, when using this ideal as a center. Blowing up .A2C at this center in a single blow up, we obtain by explicit computation: Ker() = cx − ay + a, ax − by − a ⊂ C[x, y, a, b, c].

.

In each of the three charts, the local equations are non-trivial and the exceptional divisor possesses 3 disjoint components. It is a fact that Groebner bases are of doubly exponential

24

A. Frühbis-Krüger

complexity in the number of variables. Hence this is a rather unpleasant starting point for subsequent computations. Now we try the other option of blowing up one point at a time. For the center of the first blow up, we choose the origin and obtain two charts, each of which is again an .A2 . Explicit computation of the strict transforms of the two remaining points shows that in each chart we see precisely one of them but not the respective other. So each of the two charts needs to be blown up once again at a point, which leads again to a total of four final charts, each of which is isomorphic to .A2C . In general, blowing up at irreducible centers is preferable to using fewer blow-ups at more complicated smooth centers in practice. One reason is the shortcut for blowing up at regular sequences, which will be explained in the next subsection and avoids the cost of elimination. Another advantage is that irreducible centers provide irreducible exceptional divisors (in the smooth ambient space) and hence more control on the setting after the blow-up. In a single blow-up or a very short sequence of blow-ups this is not really relevant, but for instance exploring the intersection properties of the exceptional divisors after a rather complicated resolution of singularities becomes significantly easier. The question of irreducibility of the center, however, is rather subtle in practice, as computations can only be performed over computable fields such as .Q and algebraic extensions thereof, whereas we tend to think and reason mostly over .C. So even after primary decomposition, which is not cheap by itself, we can only be sure to have .Qirreducible components; absolute primary decomposition can give an answer over .C, but at a cost that largely outweighs the benefit.

3.2

Blowing Up at a Regular Sequence

From the definition of the map . in the previous subsection, it is obvious that the number of elements of a chosen set of generators for the ideal I makes a difference for the number of new variables to be introduced. So in general it is desirable to have as few elements as possible for the given ideal. In particular using a Groebner basis of the ideal is not the right way to go! Instead, we prefer the set of generators to be as small as possible. In the minimal setting this even saves the work of computing the kernel of . by elimination: Lemma 3.7 Let U be an affine chart with coordinate ring A and .I = f1 , . . . , fm  ⊂ A an ideal. Assume in addition to this that the generators .f1 , . . . , fm of I form a regular sequence in A. Then the kernel of the map ., which was introduced above for computing the blow up in the general setting, is of the form ker() = fi yj − fj yi | 1 ≤ i < j ≤ m.

.

A Computational View on Hironaka’s Resolution of Singularities

25

What we actually see in the generators of .ker() are just the well-known Koszulrelations, which are the only relations among the .fi in this case. In each chart .D(yi ) of the blow-up this implies that the equations of .π −1 (U ) are all of the form: fj =

.

yj fi ∀j = i. yi

The simplest case of such a center I , which happens to be a complete intersection, is of course a single point. We already used this tacitly in Example 3.6 for blowing up the single points, but also saw that the blow-up became much more complicated, when blowing up at all three points at the same time—which is no longer a complete intersection. Remark 3.8 (Computational Point of View) In large and very large examples, it can be very helpful to remember that a smooth scheme is a locally complete intersection. Therefore we can always hope to cover the center by principal open sets in such a way that in each chart the corresponding ideal can be generated by a regular sequence. However, finding such a covering in practice is non-trivial. It can be understood as a very simple special case of the constructions we are going to see in the last three sections of this chapter. For an explicit discussion of this special case from the computational point of view we refer the interested reader to [2], where such a covering is used as a bridge toward massively parallel computations.

3.3

Transforms

Taking the perspective of embedded resolution of singularities of a subscheme X of a smooth scheme W , the transformation of subschemes under blow-ups will be central to our considerations. We have already recalled the notions of total and strict transforms in 3.2. Geometrically speaking, the total transform is the full preimage of X under the blow-up. The strict transform is the Zariski-closure of the set of those points in the preimage, which do not lie in the exceptional divisor; less formally, we can understand this as the union of the components of the preimage which are not contained in the exceptional divisor of the blow-up. Relevant for the embedded resolution process are also two intermediate notions of transform, the weak transform, where only the exceptional divisor itself is removed from the total transform, but not components embedded in it, and the controlled transform for which the exceptional divisor is only removed in a prescribed multiplicity. These two objects are geometrically not as obvious as the strict transform, but they are very accessible on the algebraic and computational side and their meaning for the resolution process will become clear in the second half of this chapter.

26

A. Frühbis-Krüger

Algorithmically, the starting point for computing all transforms is the total transform and the main tool is saturation by the ideal of the exceptional divisor. To this end, we again pass to the affine situation by means of a covering by charts. Lemma 3.9 Let U , I and Y be as before and denote by .IE the ideal of the excpetional divisor, then the transforms of a subscheme .Z ⊂ U with ideal .IZ can be computed as π ∗ (IZ ) = IZ OW˜ IZ˜ = (IZ OW˜ : IE ∞ ) (IZ OW˜ : IEk ) . where k = max{l ∈ N|(IZ OW˜ : IEl−1 ) = (IZ OW˜ : IEl ) · IE } controlled transform (w.r.t. a control c) (IZ OW˜ : IEc ) total transform strict transform weak transform

Remark 3.10 It is easy to see that strict and weak transform coincide for hypersurfaces. Remark 3.11 (Computational Point of View) Saturation is in general an expensive operation, as it consists of a sequence of ideal quotient operations, each of which is a Groebner basis computation with respect to an elimination ordering. However, in the computation of controlled, weak and strict transform we are in a by far less expensive special case. We saturate w.r.t. the ideal of a Cartier divisor. In practice, this saturation will often even be performed with respect to a principal ideal generated by one of the variables. Example 3.12 (Warning!) It is, in general, not possible to avoid the saturation when computing a strict transform: Let .X = V (x 2 − y 4 , x 2 − z3 ) ⊆ A3C and consider the blow-up of .A3C at the origin. In the chart where the exceptional divisor corresponds to .IE = y, the total transform of X is given by the ideal u2 y 2 − y 4 , u2 y 2 − w 3 y 3  = y 2 (u2 − y 2 ), y 2 (u2 − yw 3 ) ⊂ C[u, v, y]

.

(with the obvious substitutions .x = uy and .z = uw for the local description of the blowup). On the other hand, the ideal of the strict transform contains the polynomial .y − w 3 , as .y 3 (y − w 3 ) is in the ideal of the total transform. For the weak and controlled transform, consider a blow-up with center inside the locus of maximal order of .IX and a polynomial f generating the ideal of the exceptional divisor in this chart. Then the saturation can be understood as iterated division of the generators of .π ∗ (IZ ) by f . But this point of view does not provide an algorithmic advantage, as each ideal quotient is reduced to division of the generators by f anyway in this case (see Lemma 2.11). For the weak transform the exponent k from above corresponds to the maximal order of .IX in this setting; for the controlled transform a control c between 1 and this maximal order makes sense.

A Computational View on Hironaka’s Resolution of Singularities

27

Remark 3.13 Despite warning 3.12, the strict transform can be computed from a suitable set of generators of the ideal in the local setting at a point, more precisely using a so-called local monomial ordering and thus emulating computations in the local ring at the origin. To this end, the set of generators needs to be a standard basis of the ideal in the sense of computer algebra (see Remark 2.1). Unfortunately, computing standard bases at all points is in general not feasable, but for theoretical arguments such reasoning is very useful. We now turn to the promised example demystifying the algebraic meaning of the controlled transform: Example 3.14 Let .X = V (x 2 +y 2 z +yz2 ) ⊆ A3C (a .D4 surface singularity), consider x as main variable and .g = y 2 z + yz2 as the coefficient of the monomial .x 0 . We then proceed by a blow-up of .A3C at the origin. In this setting it makes perfect sense to ask for the strict (or equivalently the weak) transform of the hypersurface X. But also the question, how the coefficient g is transformed under the blow-up, is rather natural. Consider the chart with exceptional divisor .E = V (y): IX,total = y 2 (x 2 + yz + yz2 )

.

IX,strict = x 2 + yz + yz2  gcontrol,2 = yz + yz2 gstrict = z + z2 As the order of .IX attains its maximal value 2 precisely at the origin, we know that we need to remove the exceptional divisor with multiplicity 2 from the total transform of X to pass to the strict transform. The coefficient g, on the other hand, has itself order 3; see .gstrict for how it would be transformed when considered by itself. Keeping in mind that it is the coefficient of .x 0 , we can remove the exceptional factor from it at most twice as can be seen in .gcontrol,2 . In summary, we need to transform the coefficients by means of the controlled transform, inheriting the ‘control’, i.e. the multiplicity to extract, from the original polynomial.

3.4

Examples

To develop intuition for the use of blow-ups in resolution of singularities, it is vital to consider suitable examples beyond the ones usually covered in courses and textbooks. Let us first come back to the embedded situation. From theory, we know that we can compute the transform of some scheme .Z ⊂ W directly as blow-up of Z at some center .Y ⊂ Z ⊂ W or as the strict transform of Z under the blow-up of W at the same center. But how does the choice between the options affect the practical side of blow-up?

28

A. Frühbis-Krüger

Example 3.15 (Embedded—Abstract) Let .W = A6C and let X be the determinantal codimension 2 variety V (xv − uy, x 3 + xy 2 w − uz, x 2 y + y 3 w − vz)

.

and blow up along .I = x − v, y − v, z − v, u + v ⊂ OX . Computing the blow-up of X at this center involves the elimination of one variable from an ideal consisting of 7 generators and involving a total of 11 variables; alternatively one might first want to check algorithmically whether the given generators of I form a regular sequence for .OX and hope to use the variant without elimination. In either case, the result will be represented as a subvariety of dimension 4 in .A6C × P3C . Alternatively, we can blow up the ambient space .A6C along I and use that the given set of generators is a regular sequence in .OA6 . In this case, we can avoid elimination of C

variables and directly see that we obtain a smooth subscheme of .A6C × P3C . Computing the strict transform of the subscheme X, is a rather simple kind of elimination (as the exceptional divisor is Cartier) and leads to the same result as the significantly more involved computation before. As a third variant, it is also possible to describe the center used in the first variant as a subscheme of .A6C . This involves determining a pre-image of the .OX -ideal of the center under the epimorphism .OA6 −→ OX . In general, this involves again an elimination. C In summary, the cheapest and most straightforward computation is the second of the three alternatives, which is most often used in practice and implementations. In practice, it is often useful to perform blow-ups of a subscheme X of a smooth ambient scheme W by passing to affine charts covering W , blowing up the ambient .Ak of each of the charts and then taking the appropriate transforms of .W |U and .X|U . It may well happen, that some of the charts do not cover any point of W or of X which is outside the other charts. In this case, the chart may safely be dropped from the covering of W . An example of this aspect is the following (Please bear in mind that the fractions appearing in the example are just symbols encoding the passage to the affine charts as before.) Example 3.16 Let .W = A3C , .X = V (x1 , x22 − x32 ) and blow up at the origin. then we obtain the following three charts: Chart 1: .E = V (x1 ), .IX˜ = 1 Chart 2: .E = V (x2 ), .IX˜ =  yy12 , 1 − 2

y3 2 y2 

Chart 3: .E = V (x3 ), .IX˜ =  yy13 , yy23 − 1 Obviously Chart 1 does not see the strict transform of X at all and may hence be dropped, when passing from the embedded situation .X ⊂ W to the abstract situation of X. The only

A Computational View on Hironaka’s Resolution of Singularities

29

locus of Chart 3, which is not seen by Charts 1 and 2 is .V (y1 , y2 ). But its intersection with the strict transform is again empty, so Chart 2 suffices to cover X in this case. In an OSCAR-session this looks as follows: julia> W = affine_space(QQ,3) # affine 2-space W Spec of Multivariate Polynomial Ring in x1, x2, x3 over Rational Field julia> (x,y,z) = gens(OO(W)) 3-element Vector{QQMPolyRingElem}: x1 x2 x3

# naming the generators

julia> I = ideal(OO(W),[x,y^2-z^2]) ideal(x1, x2^2 - x3^2)

# ideal of X

julia> X = subscheme(W,I) # defining X Spec of Quotient of Multivariate Polynomial Ring in x1, x2, x3 over ... julia> J = ideal(OO(X),[x,y,z]) ideal(x1, x2, x3)

# ideal of center

julia> bl = blow_up(X,J) Blow up of a Covered Scheme with 1 Charts leading to a Covered Scheme...

julia> affine_charts(domain(bl))

# look at the charts of the transform of X

3-element Vector{AbsSpec}: Spec of Localization of Quotient of Multivariate Polynomial Ring in (s1//s0), (s2//s0), x1, x2, x3 over Rational Field by ideal(x1, x2^2 - x3^2, 1, -(s1//s0)*x3 + (s2//s0)*x2, (s1//s0)*x2 - (s2//s0)*x3, (s1//s0)^2 - (s2//s0)^2) at the multiplicative set powers of QQMPolyRingElem[1] Spec of Localization of Quotient of Multivariate Polynomial Ring in (s0//s1), (s2//s1), x1, x2, x3 over Rational Field by ideal(x1, x2^2 - x3^2, (s0//s1), (s2//s1)*x2 - x3, -(s2//s1)*x3 + x2, -(s2//s1)^2 + 1) at the multiplicative set powers of QQMPolyRingElem[1] Spec of Localization of Quotient of Multivariate Polynomial Ring in (s0//s2), (s1//s2), x1, x2, x3 over Rational Field by ideal(x1, x2^2 - x3^2, (s0//s2), -(s1//s2)*x3 + x2, (s1//s2)*x2 - x3, (s1//s2)^2 - 1) at the multiplicative set powers of QQMPolyRingElem[1]

Blow-ups by hand very often involve centers consisting of the intersection of some coordinate hyperplanes, because we tend to move a smooth center to the origin and choose appropriate coordinates. This is in general not the case in automated computations, as a heuristic for finding an appropriate coordinate change is expensive and applying it again

30

A. Frühbis-Krüger

has its costs, in particular if many objects need to be transformed. A very simple example of this kind is: Example 3.17 Let .W = A3C and let the smooth center given by the ideal .I = x1 , x22 + x32 − 1, which is just a circle of radius 1 around the origin. The given generators obviously form a regular sequence in .C[x1 , x2 , x3 ] and hence the ideal of .W˜ ⊂ A3C × P1C is .y2 x1 − y1 (x22 + x32 − 1). In the two charts, the situation is the following: Chart 1: .E = V (x1 ), .W˜ |Chart1 = V (x22 + x32 − 1 − Chart 2: .E = V (x 2 + x 2 − 1), .W˜ |Chart2 = V (x1 − 2

3

y2 y1 x1 ) y1 2 y2 (x2

⊂ A4C

+ x32 − 1)) ∼ = A3C

The first chart provides a non-trivial .W˜ , computation of transforms in this chart cannot be performed by simple variable substitution as in Chart 2. As a counterpart to this observation, chart 2 provides an exceptional divisor which is not a coordinate hyperplane. Up to now, we tacitly only allowed smooth centers. This is, however, a deliberate decision which needs to be seen in the context of the task of embedded desingularization: Given a smooth ambient scheme W and a subscheme .X ⊂ W , we are looking for a smooth ambient scheme .W˜ and a smooth subscheme .X˜ ⊂ W˜ together with a proper birational morphism .π such that .π |X provides a (strong) resolution of X as in the following diagram: π W˜ −→ W

.

∪

∪ π |X

X˜ −→ X. The following example shows what happens as soon as we allow singular centers of blowups: Example 3.18 Let .W = A3C and consider a blow-up at the simplest possible singular center .I = x1 , x2 x3 . Then .W˜ is singular, because its ideal in one of the charts is . yy21 x1 − x2 x3 , singular at .V (x1 , x2 , x3 , yy21 ). In OSCAR this looks as follows: ulia> W = affine_space(QQ,3) # affine 2-space W Spec of Multivariate Polynomial Ring in x1, x2, x3 over Rational Field julia> (x,y,z) = gens(OO(W)) 3-element Vector{QQMPolyRingElem}: x1 x2 x3

# naming the generators

A Computational View on Hironaka’s Resolution of Singularities julia> I = ideal(OO(W),[x,y*z]) ideal(x1, x2*x3)

31

# ideal of singular center

julia> bl=blow_up(W,I) Blow up of a Covered Scheme with 1 Charts leading to a Covered Scheme with 2 Charts julia> affine_charts(domain(bl)) 2-element Vector{AbsSpec}: Spec of Localization of Quotient of Multivariate Polynomial Ring in (s1//s0), x1, x2, x3 over Rational Field by ideal(-(s1//s0)*x1 + x2*x3) at the multiplicative set powers of QQMPolyRingElem[1] Spec of Localization of Quotient of Multivariate Polynomial Ring in (s0//s1), x1, x2, x3 over Rational Field by ideal((s0//s1)*x2*x3 - x1) at the multiplicative set powers of QQMPolyRingElem[1]

It is, of course, conceivable to allow centers with mild singularities and take care of the arising singularities of the ambient space later on. This, however, only changes the order in which the different subtasks are handled and usually does not simplify the overall problem, unless one is only interested in a partial solution allowing well-controlled mild singularities of the ambient scheme. To end this section, we add one remark on blowing up in the arithmetic setting, where blow-ups may be used theoretically and in implemented algorithms in precisely the same way. This relies on Groebner bases over the integers. As computed results may look surprising at first glance in this setting, we give a very simple example. Example 3.19 Consider the two-dimensional scheme .W = Spec Z[x] and the blow-up of W at the maximal ideal .3, x. This provides the following charts: Chart 1: .IE = 3, .W˜ |Chart1 = SpecZ[ yy12 ] Chart 2: .IE = x, .W˜ |Chart2 = SpecZ[x, yy22 ]/3 − x yy12 

3.5

Weighted Blowing Up

As an aside at the end of this section, we briefly touch the algorithmic side of weighted blow-ups which appear in the later chapters of this book. We deliberately state this without too much context, as the theory will be introduced later in full detail. It might be a good idea for the reader to postpone in depth reading of this subsection until after reading the later parts. For their computation of weighted and log-weighted blow-ups we need analogous considerations to the ones in the earlier parts of this section, although this might be hidden behind notation at first glance. Therefore, we briefly revisit the blow-up to adjust our point of view and then sketch the weighted case.

32

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For scheme W and a coherent ideal sheaf .I on W , we had stated the blow-up of W at I by

.

W˜ = P rojW (



.

I d t d ),

d≥0

this is to say by the P roj construction applied to its Rees-algebra. We might as well rephrase the passage to the blown-up scheme in a slightly different way: W˜ = (SpecW (



.

I d t d ) \ V (It))/Gm

d≥0

Taking out the origin and quotienting by the action of the group .Gm just encodes the transition from the affine to the projective setting here. The computation of the blow-up (of an affine chart U ) still boils down to the same construction as before: With the notation .A = OW (U ) and .I(U ) = J = f1 , . . . , fk  we need to compute the kernel of the ring homomorphism  : A[y1 , . . . , yk ] −→



.

J n t n (⊆ A[t])

n≥0

defined by .(yi ) = tfi . Introducing an inverse s for t, we can state this even more conveniently: Setting A = A[s, y1 , . . . , yk ]/syi − fi | 1 ≤ i ≤ k

.

we obtain W˜ = (Spec(A ) \ V (y))/Gm .

.

This is the perspective from which usual blow-ups can easily be seen as a particular case of weighted and log weighted blow-ups. Instead of considering a center .J = f1 , . . . , fk  for a partial regular system of parameters .f1 , . . . , fk , we are now facing a center 1 w

1 w

Iweighted = f1 1 , . . . , fk k 

.

or even 1 w

1 w

1

1

Ilog,weighted = f1 1 , . . . , fk k , m1d , . . . , mrd 

.

A Computational View on Hironaka’s Resolution of Singularities

33

where .w1 , . . . , wk , d ∈ Z and .m1 , . . . , mr are monomials. Writing .J = Ilog,weighted the adjusted map . now reads:  : A[y1 , . . . , yk , z1 , . . . , zr ] −→ A[s, t]/1 − st

.

is now defined by .(yi ) = t wi xi and .(zi ) = t d mi and A = A[s, y1 , . . . , yk , z1 , . . . , zr ]/s wi yi − fi , s d zj − mj | 1 ≤ i ≤ k, 1 ≤ j ≤ r.

.

We obtain the classical blow-up for .r = 0 and .wi = 1 for all .1 ≤ i ≤ k. From the great similarity of the construction, it is now obvious that no additional algorithmic considerations are required for weighted blow-ups. Note, however, that the .Gm -action in this more general context encodes the weights and needs to be stored as part of the data. As of this writing functionality for weighted blow-ups is not yet available in OSCAR, a proof-of-concept implementation can be found in SINGULAR in [16].

4

The Locus of Maximal Order

As already mentioned before, the order of an ideal locally at a point plays a central role as a building block of the invariant controlling the resolution process. In this section we study its properties and discuss its algorithmic computation. The more theoretical discussion in the first subsection introduces the order of an ideal and related invariants and allows us to shift the point of view from an invariant at a point to the locus of maximal invariant. For the remainder of the section we then break up the computation of the locus of maximal order into three different computational tasks: the case of trivial ambient space, the case of non-trivial ambient space in suitably chosen charts and the choice of a suitable covering to apply the previously introduced techniques.

4.1

The Order of an Ideal

For a hypersurface .X = V (f ) ⊂ AnC the order at a point is precisely the order (or multiplicity) of the power series expansion of f at the point x. It is quite intuitive that a power series of lower order should be considered ‘better’ than one of higher order. In particular a hypersurface of order 1 at x is smooth at x. But what does it depend on and how should it be generalized to the order of an ideal? Example 4.1 For this example we pass to the local situation at the origin. Consider k a hypersurface germ .(X, 0) ⊂ (An+1 C , 0) with defining ideal .IX,0 = f  = z + k−1 i i=0 ai (x)z . This f is actually a polynomial with respect to the main variable z and has coefficients in .C[[x]]. It is quite obvious that the order of f at the origin cannot exceed k

34

A. Frühbis-Krüger

due to the summand .zk . Moreover, we see that ordx (X) = k ⇐⇒ ordx (ai ) ≥ k − i ∀ 0 ≤ i < k.

.

Assuming that f attains this maximal order k at the origin, we can blow-up at this point. Direct computation shows that in the chart where the exceptional divisor locally has the ideal .z, the strict transform no longer meets the origin, whence this chart may safely be dropped. Since no particular properties concerning the different .xj have been fixed, all remaining charts show the same behaviour, which we illustrate on the chart with excpetional divisor corresponding to .x1 . Here the total transform of the ideal decomposes in the following way into strict transform and exceptional divisor: ftotal =

.

x1k · znew k +   exc.div. 

 k−1 

ai (x1 , x2,new x1 , . . . , xn,new x1 ) i=0

x1 k−i

 znew i . 

fstrict

Clearly, the situation improved, if the order of .fstrict dropped below k for all points in the intersection of the strict transform with the excpetional divisor. So again the local order of the coefficients needs to be considered. The appropriate generalization of the order of a single power series to the order of an ideal at a point can be understood as follows: The powers of the maximal ideal give rise to a filtration of the local ring. Then the order corresponds to the highest power of the maximal ideal still containing the power series or the ideal respectively. More formally, we have the following definition: Definition 4.2 Let .x ∈ X be a (not necessarily closed) point on a subscheme X of a smooth scheme W . We denote by .(R = OW,x , m, k = R/m) the local ring at .x (which is excellent and regular) and by .J ⊂ R the ideal locally defining .X at .x. Then 1. The order of X at x is defined as the order of the ideal J at the maximal ideal .m ordx (X) := ordm (J ) := sup{t ∈ N | J ⊂ mt }.

.

2. The maximal order of X is defined as maxord(X) := sup{ordx (X) | x ∈ X}

.

and the locus where it is attained is referred to as the locus of maximal order of X.

A Computational View on Hironaka’s Resolution of Singularities

35

3. The initial form of f with respect to .m is defined as inm (f ) := f mod mordm (f )+1 ∈ grm (R)

.

 where .grm (R) = i≥0 mt /mt+1 is the associated graded ring of X w.r.t. .m. 4. The initial ideal of J at .m is then defined as inm (J ) := inm (f ) | f ∈ J grm (R) .

.

The order is known to be an upper semi-continuous function (see e.g. [13] Chapter III §3 Corollary 1 p. 220) and the associated graded ring .grm (R) is isomorphic to a polynomial ring in .dimx (W ) variables over k. Computing the order locally at a point is not difficult at all, as this can be read off as the lowest order of a generator of J . Remark 4.3 (Standard Bases and .ν ∗ ) The order of J is just the lowest degree appearing in an element of .inm (J ). Asking for a finite set of generators for .in(J ) is the question of finding a standard basis G of J , which is a finite set of elements in R defined by the properties .G ⊂ J and .in(G) = in(J ) and is usually assumed to be minimal in the sense that the initial forms provide a minimal set of generators for .in(J ). Algorithmically it is computable (at any point with coordinates in a computable base field, after moving the point in question to the origin) by means of Buchberger’s algorithm w.r.t. a local ordering using Mora’s normal form instead of the Buchberger normal form. Given a standard basis .G = (f1 , . . . , fr ) for J with elements ordered such that .ordm (f1 ) ≤ ordm (f2 ) ≤ . . . ≤ ordm (fr ), we obtain a much finer invariant than the order ν ∗ (J ) = (ordm (f1 ), . . . , ordm (fr )),

.

which was first introduced by Hironaka. On the other hand, it is significantly coarser than the Hilbert-Samuel function, as we see in the simple example of the following three ideals in .C[[x, y]]: HS ord ν ∗ 2 J1 = x , xy 2 (2, 2) (1, 2, 1, 1, . . . ) . J2 = x 2 , y 2  2 (2, 2) (1, 2, 1, 0, . . . ) J3 = x 2 , y 3  2 (2, 3) (1, 2, 2, 1, 0, . . . ) Unfortunately, all these invariants depend on local data and do not directly generalize to simultaneous computation on a whole affine set or more precisely to computing strata of points at which the invariant exceeds a given value. The most accessible among them to

36

A. Frühbis-Krüger

such computations is the locus of order at least b on some affine open chart, but there are still many details to be considered. Therefore we approach this in three steps: • We first focus on the case of a trivial ambient space, which can always be reached in theory by an appropriate coordinate change, but is by far too costly in practice. • Then we study the local situation with a non-trivial smooth ambient space paying attention to the choice of an appropriate regular system of parameters and taking derivatives with respect to it. • The last step then shows how to find open coverings allowing a consistent choice of local systems of parameters on each of the charts.

4.2

Simplest Case: X ⊂ An

In the simplest setting, the idea of computing the locus of order at least b is straightforward: A power series has order at least b at the origin iff it does not contain any terms of lower degree. This, in turn, is the case in characteristic zero, precisely if any derivative of order up to .b − 1 vanishes at the origin. This idea can be transformed into an algorithm for determining the locus of order at least b—even in positive and mixed characteristic, if due care is applied. Observation 4.4 Let k be a field of characteristic zero and consider .R = k[[x1 , . . . , xn ]] with maximal ideal .m = x1 , . . . , xn  which is a regular local k-algebra with regular system of parameters .(x1 , . . . , xn ). The order of an element .f ∈ R can be determined by ordm (f ) = min{i ∈ N | ∃ a ∈ Nn : |a| = i ∧

.

∂ ∈ m}, ∂x a

where we abbreviate . ∂x∂ a := ( ∂x∂ 1 )a1 . . . ( ∂x∂ n )an . Considering now a hypersurface .V (f ) ⊂ Ank over a field of characteristic zero, we can easily deduce how to compute the locus of maximal order: The variables of the polynomial ring .OAnk = K[x1 , . . . , xn ] induce a regular system of parameters at each (geometric) point of .Ank by translation. Translation, however, does not affect the derivatives and we obtain: Lemma 4.5 Let k be a field of characteristic zero and .V (f ) ⊂ Ank some hypersurface. Then the locus of order at least b of f (or sloppily speaking of .V (f )) is precisely the ∂f vanishing locus of the ideal .b−1 (f ) := f  +  ∂x a | |a| < b At first glance, this seems like a direct generalization of the singular locus of the hypersurface, which happens to coincide with .V ((f )), but as soon as we pass to more general subschemes of .Ank , the difference becomes evident. Let .X ⊂ Ank be a subscheme

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37

defined by some ideal .I ⊂ k[x]. Then the locus of order at least b of X is the intersection of the loci of order at least b of all its elements. It is hence defined by the ideal b−1 (I ) :=

.

b−1 (f ).

f ∈I

Clearly it suffices to only consider the loci of a set of generators of I and not of all elements of I . Remark 4.6 (Computational Aspect) In contrast to the singular locus, the locus of order at least two is computationally cheaper: No Jacobian matrix needs to be build up and the combinatorially expensive computation of the minors of a large matrix is not necessary. However, this locus does not coincide with the singular locus of X, but is merely contained in it. It is the locus, at which we locally—in the sense of Zariski open sets— cannot find a smooth hypersurface containing X. From the construction of the locus of order at least b, one could get the idea of considering a stratification of X by the orders of I . This, however, introduces the complication of considering locally closed sets and does not provide much additional information, as the following example shows: Example 4.7 Let .X = V (xz, yz, x a + y b ) ⊂ AnC for some .a, b ≥ 3. Then the maximal order is 2 and the locus of order 2 is precisely the origin. The locus of order 1 is the complement of the origin in X. A vital part of the structural information on this space curve (determining, for instance, invariants like multiplicity, Milnor number etc.) lies in the last equation to which the stratification by the order is blind. Therefore, we will only focus on the locus of maximal order from now on, bearing in mind that during the resolution process each of the lower order strata will eventually show up in a maximal order stratum, as soon the maximal order of the transform of X has dropped to this value. Note also that the locus of maximal order can have a very rich structure itself, as we see immediately, when thinking about a hyperplane section of some scheme. Remark 4.8 The maximal order of an ideal may very well depend on different generators at different points of X. Consider, for instance, the union of the four lines .z ± 1, x + y, 2y ± z − 1, x − y ⊂ C[x, y, z] with ideal .x 2 − y 2 , (x + y − 1)2 − z2  ⊂ C[x, y, z]. This is clearly singular at the pairwise intersection points of the lines, as can be read off

38

A. Frühbis-Krüger

the Jacobian matrix .

 2x 2y 0 . 2x + 2y − 2 2x + 2y − 2 −2z

But the order of the ideal is constant of value 1 on all four lines as 1 (I ) = 2x, −2y, 2x + 2y − 2, −2z = 1.

.

because at the locus of order 2 of the first equation ,which is .V (x, y), the second equation has order 1 and vice versa.

4.2.1 Beyond Characteristic Zero Up to now, we had restricted our attention to characteristic zero, because of the obvious problems arising from taking derivatives in positive characteristic (see Example 4.9 below). Fortunately, it is possible to avoid this problem by slightly modifying the derivatives to be considered, namely passing to Hasse-derivations (cf. [8], sections 2.5, 2.6,). In a nutshell, they can be understood as follows: Given .f ∈ k[x1 , . . . , xn ], we pass to the larger ring .k[x1 , . . . , xn , T1 , . . . , Tm ] with new variables .T1 , . . . , Tn and consider

f (x1 + T1 , . . . , xn + Tn ) =

fa (x1 , . . . , xn )T a .

.

a∈Nn

Then the Hasse-derivative of f by .x a is the coefficient of .T a in the above expression, i.e. .

∂ (f ) = fa (x1 , . . . , xn ). ∂x a

Note that we used the same notation as for the usual derivatives in characteristic zero, because in our context there is no danger of confusion. Example 4.9 Consider .f = x 2 + y 2 ∈ F2 [x, y]. Clearly all “usual” derivatives of f vanish. But we have f (x + T1 , y + T2 ) = x 2 + T12 + y 2 + T22 ∈ F2 [x, y, T1 , T2 ],

.

whence only the first derivations and the mixed one vanish, whereas ∂ .

∂x (2,0)

(f ) = 1 =

∂ ∂x (0,2)

(f ).

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39

Using Hasse-derivatives, all considerations of this subsection immediately carry over to positive characteristic. This is not the point where the insurmountable obstacle in positive characteristic arises! Even in mixed characteristic, a similar approach to computing the locus of maximal order is feasable, but requires some rather unusual perspective of taking ‘derivatives by prime numbers’ to deal with polynomials like .12 − xy ⊂ Z[x, y] of which the locus of maximal order 2 is precisely given by the ideal .2, x, y. For more details on the computational aspects of this case, we refer the reader to [7], section 3. In what follows, we concentrate on the characteristic zero setting for simplicity of presentation.

4.3

General Case I: Local Situation at p

A crucial point in the previous discussion was the use of derivations. To this end, we used the local regular system of parameters induced by the variables of the coordinate ring of n .A via translation. Now that we are no longer dealing with a trivial ambient space, the task of determining such a regular system of parameters arises. In general, it is a highly non-trivial task to find such a system of parameters, but we are not in a generic setting. We have the advantage that all data is already represented on the computer by means of ideals in polynomial rings. In other words, our smooth ambient space W is itself embedded in some .AN . This provides us with a local system of parameters of .AN from which we can select a suitable system of parameters for our ambient space W . For providing an algorithmic approach to this, we start by a more detailed (but still theoretical) local description. Fixing a point .p ∈ W , we already know from our setting, that .OW,p is a regular local ring, say of dimension d and with maximal ideal .m and we need to find a minimal set of generators .X1 , . . . , Xd for .m. By the Cohen structure theorem, we additionally know that there is an isomorphism of local rings   : k[[y1 , . . . , yd ]] −→ O W,p

.

yi −→ Xi .  By means of this isomorphism, we can define derivatives of some element .f ∈ O W,p w.r.t. the given system of parameters as .

  ∂ ∂ −1  ( (f ) ∈O (f ) :=  W,p . ∂y a ∂Xa

40

A. Frühbis-Krüger

As W is smooth at p, it is a complete intersection in a sufficiently small neighbourhood of p. So for the theoretical discussion, we may assume that the ideal .IW,p ⊂ OAN ,p of W is generated by some elements .f1 , . . . , fN −d ∈ OAN ,p which form a regular sequence. By smoothness, we further know that at least one maximal minor of the Jacobian matrix  M of .f1 , . . . , fN −d , say h, will be a unit in .O W,p ; without loss of generality we assume h to arise from the last .N − d columns. Then we can obviously choose the remaining variables .x1 , . . . , xd ∈ OAN ,p as our regular system of parameters for W at p. Computing derivatives or more precisely .(f ) ∈ OW,p for some .f ∈ OAN ,p , however, requires even more explicit considerations. Denote the matrix of cofactors of the invertible .(N − d) × (N − d) submatrix of the Jacobian matrix by A and observe that .

1 A · M = (∗ | EN −d ) h

where .EN −d denotes the unit matrix and .∗ stands for some matrix of size .(N − d) × d. Avoiding denominators, we can also write A · M = (∗ | h · EN −d )

.

(with an appropriate .∗ of course). Performing these row operations on the Jacobian matrix can be mimicked on the level of the generators of .IW,p applying A to the vector T .(f1 , . . . , fn−r ) to obtain (f˜1 , . . . , f˜N −d )T =

.

1 A · (f1 , . . . , fN −d )T . h

It is easy to see that .IW,p = f˜1 , . . . , f˜N −d . Moreover, the new system of generators gives rise to a Jacobian matrix of the desired structure .(∗ | h · EN −d ), when viewing the entries as elements of .OW,p (Use the chain rule and scratch all contributions in .IW,p ). Now consider the preimage of the ideal .f  = h · f  ⊂ OW,p under the canonical projection from .OAN ,p and follow the same reasoning to obtain a matrix   ∗  | h · EN −d ∗ | h · EN −d   . ∼ ) ∂(h·f ) . . . ∂(h·f H1 . . . Hd | 01×(N −d) ∂x1 ∂xN Then .(f ) = f, H1 , . . . , Hd  ⊂ OW,p .

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41

Example 4.10 Consider .f = x 3 + y 3 + z2 and .f  ⊂ C{x, y, z}/z − y 2 . Then above matrix reads    0 −2y |1 0 −2y |1 0 −2y | 1 . ∼ ∼ 3x 2 3y 2 | 2z 3x 2 3y 2 + 4yz | 0 3x 2 3y 2 + 4y 3 | 0 Therefore, .(f ) = x 3 + y 3 + z2 , 3x 2 , 3y 2 + 4y 3 . Now iterate this to obtain .2 (f ) = x 3 + y 3 + z2 , 3x 2 , 3y 2 + 4y 3 , 6x, 6y + 12y 2  and eventually .3 (f ) = 1, whence the order of f in W at p is 3.

4.4

General Case II: Covering an Affine Chart

Knowing, how to compute the locus of maximal order locally at a point for some .f ∈ OW,p immediately implies that we can also compute it for any ideal in this ring. It does, however, not yet provide us with the subscheme of W at which .f ∈ OW,p (or more generally some given ideal) has maximal order. To determine this, we need to find a covering of W such that the regular system of parameters (inherited from the ambient .AN p ) is induced by a fixed set of variables at each point of a given patch. As the local construction provides polynomial data as long as the input is polynomial, the above computation can then be performed on the whole patch. The results on all patches need to be glued in the end to provide the desired locus.

4.4.1 How to Choose a Covering The choice of an appropriate covering is already hinted in the previous construction: We choose a covering such that on a given patch the same minor of the Jacobian matrix does not vanish. By the smoothness of W , this yields the covering: 

W =

.

(D(m) ∩ W )

m∈minors(M,n−D)

where M denotes the Jacobian matrix of a set of generators of .IW,p and .D(m) the complement of the hypersurface .V (m) in .AN . As the number of minors can easily become quite large, it suffices in most cases to only use some of the minors, say .{m1 , . . . , ms } such that W is covered. More technically speaking, we want to satisfy the condition 1∈

.

 IW + m1 , . . . , ms ,

which can algorithmically be tested by a radical membership test.

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4.4.2 Recombining Charts Even after reduction of the number of charts, it is still not desirable to continue with this covering, if recombination is possible. The large overlaps of the charts can lead to a significant amount of multiple storage of data. Moreover, the locus of maximal order is known to be a subscheme of .W ⊂ AN , whence it can and should be described globally. To find this description, we first observe that all constructions of this section lead to N polynomial output upon polynomial input data. So on each patch we obtain an ideal in .OA whose restriction to a chart .D(m)∩W describes the locus of maximal order on this scheme. But it may well contain components inside .V (m) which are artefacts of our computation and do not hold valid data. These artefacts can be dropped by saturation of the ideal by the principal ideal .m. After this, the data is reliable and we are free to take the union over the subschemes of .AN arising from the individual patches. Remark 4.11 (Computational Aspect) It may seem strange to apply computationally expensive Groebner basis techniques like radical membership test and saturation to keep the number of charts low. The saturations are taken w.r.t. hypersurfaces, which significantly reduces their computational costs as compared to the general case. The radical membership test on the other hand, can be expensive, but in general saves huge amounts of computations on the patches without new information. Another benefit of this recombination is that the covering does not force subsequent blow-up steps (which can be expected to be numerous) to be performed in charts, which again is a significant reduction of work.

4.5

Examples

In this subsection, we give three explicit examples to illustrate the considerations explained above: Example 4.12 In .A4C , consider the cylinder .V (g) = V (x 2 +y 2 −1) over a circle in the xyplane as the ambient space W . To compute the locus of maximal order of .f = z4 −y 2 w 2 ∈ OW we first determine a suitable covering by principal open sets. The Jacobian matrix of g is .

  2x 2y 0 0

and obviously .(D(y) ∩ W ) ∪ (D(x) ∩ W ) = W , as .V (x, y) ⊂ W . On the other hand, we cannot drop one of the charts, as this would hide the points .(1, 0, z, w) and .(0, 1, z, w) respectively from our view

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43

On .D(x), our regular system of parameters arises from .{y, z, w} and we observe that the matrix  2x 2y 0 0 . 0 2yw 2 4z3 2y 2 w does not require any preceding matrix operation before reading off (f ) = z4 − y 2 w 2 , yw 2 , z3 , y 2 w = yw 2 , z3 , y 2 w.

.

Iterating this computation we further obtain 2 (f ) = w 2 , yw, z2 , y 2 

.

3 (f ) = w, y, z 4 (f ) = 1 Hence the maximal order is 4 and this is attained at the two points .V (y, z, w, x 2 − 1) on W. On .D(y), we use the regular system of parameters .{x, z, w} and need a matrix operation to read off the derivatives:   2x 2y 0 0 2x 2y 0 0 . ∼ . 0 2yw 2 4z3 2y 2 w −2xw2 0 4z3 2y 2 w In this way we obtain (f ) = xw 2 , z3 , y 2 w = xw 2 , z3 , w

.

2 (f ) = 1 as y is a unit on .D(y). Thus the maximal order on this chart is only two, which is lower than in the other chart. Hence this chart does not contribute to the locus of maximal order. Example 4.13 Now consider the 2-sphere .W = V (x 2 + y 2 + z2 − 1) ⊂ A3C and compute the locus of maximal order of .f = xyz on W.

44

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Similar to the previous example, we use a covering by the three charts .D(x), .D(y) and .D(z); obviously the only point of .A3C , which does not lie in W , is the origin. By the symmetry of the situation, we only consider one chart explicitly: On .D(x) we see   2x 2y 2z 2x 2y 2z . ∼ yz xz xy 0 x 2 z − y 2 z x 2 y − yz2 and hence by iterating this (f ) = yz, x 2 z − y 2 z, x 2 y − yz2 

.

2 (f ) = yz, x 2 z − y 2 z, x 2 y − yz2 , x 2 − y 2 − 2z2 , x 2 − 2y 2 − z2 , y 2 − z2  Considered as ideal in .OW the latter ideal is the whole ring, as can be checked by explicit computation (slightly lengthy by hand, but quickly done in Oscar). Therefore the maximal order of f on W is 2 attained at .V (x 2 − 1, y, z) on this chart. By symmetry, we thus know that our locus of maximal order contains the 6 points .(±1, 0, 0), .(0, ±1, 0) and .(0, 0, ±1). For the last example, we only consider a covering for an ideal with more than one generator, but do not compute a loci of maximal order: Example 4.14 Consider .W = V (x 2 +y 2 −4, z2 −x) ⊂ A3C . The corresponding Jacobian matrix is  2x 2y 0 . 1 0 2z Being in codimension 2, we need to find the covering based on the 2-minors .{2y, 4xz, 0}. Neither .V (x, y) nor .V (y, z) meet W , whence we can use the covering by the two charts .D(y) and .D(xz).

5

Maximal Contact and Coefficient Ideals

A crucial step in the choice of center is the descent in ambient dimension by means of a hypersurface of maximal contact and the computation of the appropriate coefficient ideal. Before going into the details of this construction in the general case, we treat the easiest case of a single hypersurface and of an obvious choice of main variable. Subsequently, we discuss what conditions to impose for a suitable choice of main variable. This gives rise to

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45

the definition of a hypersurface of maximal contact and to an algorithmic way to determine one. Toward the end of this section, we discuss the coefficient ideal, which provides the last major ingredient for Hironaka’s descending induction on dimension. For a more theoretical textbook reference, we refer the interested reader to the books [15] or [6].

5.1

Motivation: A Special Case

To illustrate the general philosophy of this section, let us come back to the example at the beginning of the last section: f = zk +

k−1

.

ai (x)zi ⊂ C{x}[z].

i=0

This choice is not as particular as it might seem at first glance. It is a Weierstraß polynomial. So any power series of order at least 1, which happens to be z-general w.r.t. some variable z, can be transformed to this form up to a unit by the Weierstraß Preparation Theorem. We had seen that the order of f at the origin can be read off without computation: it is ord0 (f ) = min({k} ∪ {ord0 (ai ) + i | 0 ≤ i < k}).

.

In particular, the order of f at the origin attains its maximal value k, if and only if .ordo (ai ) ≥ k − i for all .0 ≤ i ≤ k − 1. This is the intersection of k (Zariski closed) loci. Handling each of these k conditions separately is not desirable for a recursive approach to the choice of center. We need a single ideal encoding this information, which we obtain by ord0 (f ) = k ⇐⇒ ord0 (ai ) ≥ k − i ∀ 0 ≤ i < k

.

k!

⇐⇒ ord0 (aik−i | 0 ≤ i < k) ≥ k!. The price for obtaining a single condition is the huge increase in order. In theory, it is possible to use fractional exponents for the ideal, but in practice this only transports the combinatorial complexity of creating this ideal to another part of the computation. If the order of f is indeed k, the hypersurface .V (z) corresponding to the main variable z is an example of a hypersurface of maximal contact; the ideal k!

Inew = aik−i | 0 ≤ i < k

.

is an example of a coefficient ideal.

46

A. Frühbis-Krüger

Now we fix such an f of order k at the origin and revisit the blow-up at the origin. In the chart in which the exceptional divisor is .V (z), the strict transform of f will not meet the origin. We will thus focus our discussion on some chart with exceptional divisor .V (xj ). There the strict transform reads: k fstrict = znew +

k−1

.

k−i a˜i znew

i=0

where .a˜i denotes the controlled transform of .ai with control i. This control is easily seen to be the correct one, as k is the maximal power of .xj which we can extract from .ftotal in k−i . this chart and the original factor .zk−i gives rise to the new factor .xjk−i znew k Considering the order of .fstrict at the origin of this chart the term .znew is a term of lowest degree in .fstrict if and only if the order of .fstrict is still k. Otherwise the order of k!

I2 = a˜i k−i | 0 ≤ i < k

.

has dropped below .k!, as one of the orders of the .a˜i dropped below its respective threshold. If the order has not dropped, we can still use .znew as the new main variable or in other words, the strict transform .V (znew ) of our chosen hypersurface is still a good choice. If the order has dropped, we need to find a new good choice of a main variable. This illustration of the philosophy already reveals the general strategy: Induce order reduction of a given ideal I by achieving order reduction of its coefficient ideal w.r.t. a hypersurface of maximal contact. However, the coefficient ideal need not be resolved completely, it only needs to drop below the threshold of .(ordm (I ))!. Moreover, we have seen that the coefficient ideal should not be transformed by strict or weak transform, but by a controlled transform respecting the control .(ordm (I ))!. This last idea gives rise to the notion of idealistic exponent or marked ideal, an ideal with a marking indicating the relevant threshold. These considerations immediately raise several questions, to be addressed in the rest of this section: • What characterizes a suitable choice of hypersurface of maximal contact? • Does such a hypersurface always exist? Can it be found algorithmically, if it exists? • If different suitable hypersurfaces exist, how do different choices affect the invariant and hence the choice of upcoming center? • How should the coefficients w.r.t. such a main variable be collected for an ideal (and not just a single polynomial/power series)? • How does such a coefficient ideal transform under blow-ups? How does the transform relate to a newly computed coefficient ideal w.r.t. the transformed hypersurface of maximal contact?

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5.2

47

Hypersurfaces of Maximal Contact

Our overall goal here is to make the choice of a hypersurface H algorithmic such that for given ambient space W and ideal I of maximal order b we have the following equivalence: Order reduction for I ∈ OW . ⇐⇒ Order reduction for suitable Inew ∈ OH with marking b!. Note that the discussion of .Inew will be postponed for a moment and we focus here on the choice of H with the partial goal of compatibility of the hypersurface with sequences of blow-ups at centers inside the maximal order locus. Example 5.1 Consider .W = A3C and .I = x 6 + y 3 − z2  and 4 different hypersurfaces: H1 = V (z)

.

H2 = V (z − x 2 ) H3 = V (z − x 3 ) H4 = V (z − x 5 ). For each of the hypersurfaces, we now follow a sequence of blow-ups suggested by order reduction of .IX∩Hi below the threshold 2. At the same time we track the effects of the blow-ups at the chosen centers for I . For simplicity we only track one path through the tree of blow-up charts, as this suffices to illustrate the arising problems. Note that the maximal order of I is 2 and is attained precisely at the origin. H1 : Starting with .IH1 = z and .Inew = x 6 + y 3 , we first blow up at the origin and obtain in the chart with exceptional divisor .V (x):

.

Istrict = x(x 3 + y13 ) − z12  IH1 ,strict = z1  Inew,control = x · (x 3 + y13 ) .

The maximal order of I and .Inew has not dropped under this blow-up, but some improvement can be guessed by the lowering of the largest exponent of x.

48

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Blowing up at the origin again, which is obviously still inside the locus of maximal order, and passing again to the chart with exceptional divisor .V (x), we then have Istrict = x 2 (1 + y23 ) − z22  IH1 ,strict = z2  Inew,control = x 2 · (1 + y23 ) .

Now the maximal order of the non-exceptional factor of .Inew,weak has dropped to one, attained at the third roots of unity. .V (z2 ) is still a good choice for the hypersurface of maximal contact and the ‘worst’ locus is now at the 3 points .V (x, −ζ, z2 ) with .ζ being the 3-rd roots of unity. .H2 : Starting with .IH2 = V (z − x 2 ) and using the same suitable centers and same charts as before, the transform of I is, of course, unchanged and we only list the transforms of .IH2 and the corresponding .Inew = x 6 + y 3 − x 4 . .IH2 ,strict = z1 − x Inew,control = x(x 3 + y13 − x)

Using the same second blow up as before, the difference becomes even more obvious, as then .IH2 ,strict = z2 − 1, which does no longer meet the locus of maximal order which sits inside .V (z2 ). .H3 : Starting with .IH3 = V (z − x 3 ), we see .I = (z − x 3 ) · (z + x 3 ) + y 3  leading to 3 3 .Inew = y  and only one obvious choice of center arising from it: .IC = z − x , y. But this contains points of non-maximal order. .H4 : Using .IH4 = z − x 5 , we have .Inew = x 6 + y 3 − x 1 0 and obtain by the blow ups: IH4 ,strict = z1 − x 4  Inew,control = x · (x 3 + y13 − x 7 ) .

and IH4 ,strict = z1 − x 3  Inew,control = x 2 · (1 + y13 − x 4 ) .

As with the first hypersurface the order of the non-excpetional factor of .Inew has now dropped to one attained at .V (1 + y13 − x 4 ). Obviously, .H1 and .H4 are suitable for our task, but not the other two hypersurfaces. As we see from .H2 , we cannot expect suitable centers to arise from hypersurfaces, for which the transforms do no longer contain the transforms of the locus of maximal order after a while. In the case of .H3 , the intersection with .V (I ) brings more structure into play than is present in .V (I ) itself.

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49

Looking at the locus of maximal order of I and the chosen hypersurfaces more closely, we see that only .IH1 and .IH4 are contained in .1 (I ), the ideal of the locus of maximal order. Definition 5.2 (Hypersurface of Maximal Contact for I ) Let I be an ideal sheaf on a smooth ambient space W . A hypersurface of maximal contact at a point p of order b is a smooth hypersurface which locally contains the locus of maximal order b in a suitable neighbourhood of p and continues to do so6 after any sequence of blow-ups at smooth centers lying inside of the locus of order b, until it drops. Remark 5.3 In the above definition, we rely on the Zariski upper semicontinuity of the order to ensure the existence of a neighbourhood of p on which b is the maximal order and on the infinitessimal upper semicontinuity to ensure that the order does not increase under blow up. Warning 5.4 This definition will be refined w.r.t. the presence of exceptional divisors as 6.5 in the last section of this chapter. Lemma 5.5 Let I be an ideal sheaf on a smooth ambient space W with maximal order b and let p be a point at which this order is attained. Then any .h ∈ b−1 (I ) with .ordmp (h) = 1 is a hypersurface of maximal contact in a suitable open neighbourhood of p. This statement feels a bit like magic: we can directly compute something on spot which knows about future blow-ups. Idea of Proof As this is a statement locally at a point p, as W is smooth and as we assume a center of a blow-up to be smooth, we can choose local coordinates such that the center is .x1 , . . . , xr  for some .r ≤ dim(W ). This can be extended to a local system of parameters .x1 , . . . , xdim(W ) and we denote the new coordinates after blow-up by .yi as in the section on blow-ups. In this simplified set-up, we can directly verify by explicit computation, how derivatives and controlled transform interact. In the chart with exceptional divisor .V (xr ), we see   ∂f ∂f(control,b) ∀i > r = . ∂xi ∂xi (control,b−1)   ∂f ∂f(control,b) = xr · ∀i < r ∂xi (control,b−1) ∂yi

6 Of course, considering the corresponding transforms.

50

A. Frühbis-Krüger



∂f ∂xr

 = xr · (control,b−1)

−

i 1, we have more than one independent choice for the linear part of the hypersurface of maximal contact at p. This also implies that we already know at least one choice for the subsequent hypersurface of maximal contact. Hence we can perform .dimk ((b−1 (I ))mp /m2p ) steps simultaneously. The first improvement was already implemented in the old implementation resolve.lib in Singular, the second one is still on the TODO-list as of this writing.

6

Exceptional Divisors and Blow-Up History

It surely did not escape the attention of the reader that up to this point there was no effort to take exceptional divisors into account. The reason for this is twofold: on one hand, the constructions are sufficiently complicated even without consideration of exceptional divisors and, on the other hand, the considerations up to now are also relevant for the algorithms in the other chapters of this book, whereas the role of exceptional divisors is completely different in the weighted resolution algorithm. For completeness of exposition of Hironaka-style resolution of singularities we cover these constructions and end the section by stating the structure of the full invariant of Hironaka-style resolution.

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6.1

A. Frühbis-Krüger

Motivation by Examples

We had already seen that the general philosophy is to always blow up at the worst locus— with a very careful notion of worst locus as introduced in the previous sections. Moreover, we had observed in examples at the beginning that exceptional divisors sometimes need to be used to store blow-up history which can in turn mark certain components. Think of Example 1.13 where the first blow-up gave rise to two copies of the original situation in two different charts, but in each of them precisely one of the two lines of the locus of maximal order was contained in the excpetional divisor. In general, many excpetional divisors arise over time, the order at different levels of coefficient ideals may have dropped in the meantime and history information in older excpetional divisors may have become outdated or even misleading. Hence the resolution process needs to decide which exceptional divisors are trustworthy and which are outdated. The criterion for this is whether the exceptional divisor arose before or after the last drop in order of the currently studied ideal. Last but not least, the further goal of attaining simple normal crossing leads to additional complications. Let us look at a sequence of blow-ups where an older exceptional divisor becomes outdated: Example 6.1 Let us follow through the first two blow-ups of a resolution, restricting our considerations to just one chart. The chosen example here is not completely trivial so that we can nicely see the maximal order dropping: Consider .V (y 5 − x 8 , xz4 − y 5 ) ⊂ A3C which can easily be checked to be of maximal order 5 precisely at the origin. Blowing up at this point and passing to the chart with exceptional divisor .V (x), we then see the weak transform Iweak = x 3 − y 5 , z4 − y 5 .

.

This ideal attains its maximal order 3 precisely at the origin of this chart. In particular the order has dropped and it is possible to choose .V (x) as new hypersurface of maximal contact, which precisely coincides with the exceptional divisor from the previous blow up. So there is no need to discuss whether this exceptional divisor’s information is obsolete. The coefficient ideal can easily be computed to be of order 8 here and to give rise to the hypersurface of maximal contact .V (x, z) ⊂ V (x). For the second coefficient ideal, the generator is a pure power of y, whence the upcoming center is again the origin. Blowing up at this point and passing to the chart with exceptional divisor .V (y), we see IE1 ,strict = x1 

.

IE2 = y

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Iweak = x13 − y 2 , z14 y − y 2  Istrict = y − z14 , x13 − z18  The weak and the strict transform differ even in maximal order here. The order of the weak transform dropped to 2, a new hypersurface of maximal contact needs to be chosen and both the transform of old exceptional divisor .E1 and the new exceptional divisor are untrustworthy, as they were generated before or upon the last decrease of maximal order. The fact that some exceptional divisor coincided with the hypersurface of maximal contact in the above example is pure luck. There is no reason why this should happen: Example 6.2 The locus of maximal order of the ideal x 2 − y 3 , z2 v − w 2 u ⊂ OA6

.

C

is the two-dimensional locus .V (x, y, z, w), which we choose as a center for the upcoming blowing up. In the chart with exceptional divisor .V (y), we see a weak transform Iweak = x 2 − y, z2 v − w 2 u

.

Clearly, any hypersurface of maximal contact arises from the order 1 generator or a linear combination thereof with the second one. In any case, there will be tangency between the hypersurface of maximal contact and the exceptional divisor. In the previous example, one goal should be to blow-up the (transform) of the locus of tangency as soon as possible and in the meantime make sure that centers are chosen within this locus. Blowing up the locus of tangency immediately does not make sense given the second generator which does not depend on the variable involved in the locus of tangency. But even if all intersections are pairwise transverse, there might be problems: Example 6.3 Consider the following setting which might have arisen from two previous blow-ups: I = x + y − z3 , y 3 − z4 ; E1 = V (x),

.

E2 = V (y)

Clearly a hypersurface of maximal contact needs to arise from the first generator and will meet each of the exceptional divisors transversely. But the situation on .V (I ) is not normal crossing: on such a hypersurface .V (x + y − z3 + α · (y 3 − z4 )) the intersections of the excpetional divisors are not transverse to each other, as we obtain .V (y − z3 + α · (y 3 − z4 ), x) ∩ V (x − z3 + α · (−z4 ), y) = V (x, y, −z3 − αz4 ).

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Example 6.4 Consider, on the other hand, .W = A2C and I = x 2 + y 2 − 1,

.

E1 = V (x − 1), E2 = V (y + 1).

Obviously .V (I ) is the only candidate for a hypersurface of maximal contact for .V (I ), but both exceptional divisors are tangent to it. However, the intersections of the exceptional divisors with .V (I ) meet transversally in a trivial way, as they do not meet at all. So on .V (I ), they are simple normal crossing, but they are not simple normal crossing with .V (I ). These considerations motivate the definition of a hypersurface of maximal contact in the presence of exceptional divisors: Definition 6.5 (Hypersurface of Maximal Contact for I and E) Let I be an ideal sheaf on a smooth ambient space W , let .E = {E1 , . . . , Es } be a collection of exceptional divisors on W and let b be the maximal order of I . A hypersurface of maximal contact is a smooth hypersurface H which • locally at p contains the locus of maximal order • continues to do so after any sequence of blow-ups at smooth centers lying inside of the locus of this order • has transversal intersections with the exceptional divisors in a neighbourhood of p • satisfies: .E ∩ H has transversal intersections in H locally at p

6.2

The Full Resolution Invariant

We are now ready to plug all items into the resolution process. To this end, we will follow the algorithmic variant of [3]. A convenient way to phrase the resolution process is by a resolution invariant whose maximal locus always selects the upcoming center. The overall structure of the invariant at any given point p is the following: (invd ; invd−1 ; . . . ; inv2 )

.

where the ‘;’ after .invi+1 marks the computation of a coefficient ideal .Ii and .invi denotes the contribution to the invariant arising from .Ii . On each level, the invariant .invi is itself of the form invi = (ordp (Ii ), nEi (p)),

.

where .ordp (Ii ) is the order of the ideal .Ii at p and .Ei is the collection of exceptional divisors created after the order of .Ii+1 dropped the last time, but before the one of .Ii dropped. .nEi (p) is the number of irreducible components of .Ei containing p.

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Remark 6.6 (Computational Aspect) In the formulation of the invariant, the local value nEi (p) is easy to phrase, but for the algorithmic treatment we again need a formulation of the respective locus. To this end, we need to modify the marked ideal .(I, b) representing the locus of maximal order b by intersecting its locus with the locus of .nEi -fold intersections of the relevant exceptional divisors. More precisely, we pass to the companion ideal .Inew = I + J b , where

.

J =

n

Ei 

.

I (Hαi )

α⊂{1,...,#Ei } i=1 #α=nEi

and .Hαi denotes the exceptional hypersurface at position .αi in .Ei . The subsequent hypersurface of maximal contact and coefficient ideal are then computed for .(Inew , b) which still has maximal order b as the order of .J b is at least b. Remark 6.7 (Villamayor’s Monomial Case) Up to this point, we used a simplified view on the handling of the coefficient ideal: we assumed that we always consider the controlled transform. However, the controlled transform is a product of the weak transform and an exceptional factor and improvements to the weak transform need priority over the improvements to the exceptional factor. To achieve this, Villamayor’s approach distinguishes between three situations. To describe them, we denote by c the control, which we obtained when forming .Icontrol , the controlled transform of the coefficient ideal, and .Iweak , its weak transform. • .ordp (Iweak ) > c The choice of center continues by order reduction of the weak transform as the next step. • .0 < ordp (Iweak ) ≤ c As the order of .Iweak is already below c, exceptional factors need to be considered in addition to it, to achieve suitable order reduction of the controlled transform. More ordp (Iweak ) c + Icontrol precisely, the upcoming choice of center aims at order reduction of .Iweak below the value of .c + ordp (Iweak ). • .ordp (Iweak ) = 0 A purely combinatorial choice of center, Villamayor’s monomial case (explained in section 20 of [3]), can be performed in this setting. Remark 6.8 (Bierstone-Milman and the Hilbert-Samuel Function) The approach of Bierstone and Milman [1] follows the same train of thought of making Hironaka’s approach algorithmic and can also be rephrased by means of an appropriate resolution invariant. They start from the maximal locus of the Hilbert-Samuel function, use the order of coefficient ideals and take exceptional divisors into account.

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In comparison to the approach of Villamayor and his collaborators, the locus of the Hilbert-Samuel function gives a more precise entry point, which poses more computational problems. On the other hand, Bierstone’s and Milman’s approach tends to need fewer blow-ups and hence fewer charts. We had already seen that the existence of hypersurfaces of maximal contact breaks down in positive characteristic. Hauser tried to overcome this problem by allowing the passage from a chosen hypersurface to another one maximizing the order of the upcoming coefficient ideal, whenever needed. This approach, in turn, runs into another problem in positive characteristic: the breakdown of infinitessimal semicontinuity of the order of the coefficient ideal. The next example illustrates this nicely, but could not be phrased earlier as it requires the presence of at least 2 exceptional divisors before the blow-up. Example 6.9 (Hauser’s Kangaroo Points in Positive Characteristic [11]) Consider .X = V (x 2 + yz(y 2 + z2 + w 5 )) ⊂ A4F2 with exceptional divisors .E1 = V (y) and .E2 = V (z). Using .V (x) as hypersurface of maximal contact, we see that the coefficient ideal is of the form .yz(y 2 + z2 ) and hence of order 2 after cutting away the exceptional factors. Blowing up at the origin and passing to the chart with new exceptional divisor .V (y), we obtain the following strict transform Istrict = x12 + y 4 z1 (1 + z12 + y 3 w15 )

.

Now we consider the situation locally around the point .(0, 0, 1, 0) by means of a coordinate change .z2 = z1 + 1 and observe Istrict = x12 + y 4 · (z2 + 1) · (z22 + y 3 w15 ).

.

But .y 4 z22 is a square and because of the characteristic 2, we have .(x1 +y 2 z2 )2 = x12 +y 4 z22 . Choosing as new main variable .x2 = x1 + y 2 z2 , we finally obtain Istrict = x22 + y 4 z23 + h.o.t.

.

which leads to a coefficient ideal of order 3 after cutting away the exceptional divisors. So the order of the coefficient ideal increased in this example. Remark 6.10 (CJS as a Simplified Surface Variant) In the case of surfaces, it is possible to completely avoid the use of hypersurfaces of maximal contact and coefficient ideals. The vital information arises from the maximal locus of the Hilbert-Samuel function and a different handling of exceptional divisors, organised by the moment of birth of the individual divisors and one additional property phrased as ‘dominating the center’. This approach is independent of the characteristic and has been introduced by Cossart, Jannsen

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and Saito in [4]. For algorithmic considerations in mixed characteristic, we refer the interested reader to [7]. An approach which can completely avoid the use of exceptional divisors is the weighted resolution, which will be covered in much of the rest of this volume.

References 1. E. Bierstone, P. Milman, Uniformization of analytic spaces. J. Am. Math. Soc. 2, 801–836 (1989) 2. J. Boehm, W. Decker, A. Fr"uhbis-Krüger, F.-J. Pfreundt, M. Rahn, L. Ristau, Towards massively parallel computations in algebraic geometry. Found. Comput. Math. 21(3), 767–806 (2021) 3. A. Bravo, S. Encinas, O. Villamayor, A simplified proof of desingularization and applications. Rev. Mat. Iberoamericana 21(2), 349–458 (2005) 4. V. Cossart, U. Jannsen, S. Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes. ArXiv e-prints, February 2013 5. D. Cox, J. Little, D. O’Shea, Ideals, Varieties and Algorithms (Springer, New York, 1996) 6. S. Cutkosky, Resolution of Singularities (American Mathematical Society, Providence, 2004) 7. A. Frühbis-Krüger, L. Ristau, B. Schober, Embedded desingularization for arithmetic surfaces toward a parallel implementation. Math. Comput. 90, 1957–1997 (2021) 8. J. Giraud, Contact maximal en caractéristique positive. Ann. Sci. ENS, 4eme serie 8, 201–234 (1975) 9. G.-M. Greuel, G. Pfister, A Singular Introduction to Commutative Algebra (Springer, Berlin, 2008) 10. R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, No. 52 (Springer, New York, 1977) 11. H. Hauser, On the problem of resolution of singularities in positive characteristic. Bull. Am. Math. Soc. (N.S.) 47(1), 1–30 (2010) 12. H. Hauser, J. Lipman, F. Oort, A. Quirós (eds.), Resolution of Singularities (Obergurgl 1997) (Birkhäuser, Basel, 2000) 13. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964); ibid. (2) 79, 205–326 (1964) 14. H. Jung, Darstellung der funktionen eines algebraischen körpers zweier unabhängigen veränderlichen x,y in der umgebung x=a, y= b. J. Reine Angew. Math. 133, 289–314 (1908) 15. J. Kollár, Lectures on Resolution of Singularities, vol. 166. Annals of Mathematics Studies (Princeton University Press, Princeton, 2007) 16. J. Lee, A. Frühbis Krüger, D. Abramovich, RESWEIGHTED: a SINGULAR 4-1-2 library for computing weighted resolution of singularities (2020) 17. J. Lipman, Desingularization of two-dimensional schemes. Ann. Math. (2) 107(1), 151–207 (1978) 18. T. Mora, La queste del saint gra(al). Disc. Appl. Math. 33, 161–190 (1991) 19. R. Narasimhan, Hyperplanarity of the equimultiple locus. Proc. AMS 87(3), 403–408 (1983) 20. The OSCAR development team. OSCAR — Open Source Computer Algebra Research system (2022). https://oscar.computeralgebra.de 21. O. Villamayor, Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989)

Stacks for Everyone Who Cares About Varieties and Singularities Dan Abramovich

1

Introduction

1.1

Statement of Purpose

Algebraic geometry is about the geometry of VARIETIES. Anything else is meant to help in understanding the geometry of varieties, and must be measured by that yardstick. So, for instance, we can all agree that the entire theory of schemes comes to explain degenerations, deformations, and rational points of VARIETIES.

The purpose here is to introduce algebraic stacks, specifically those of Deligne and Mumford, as a tool in problems of singularities—of VARIETIES, of course. On a simple-minded level, smooth stacks are a sneaky way to introduce hidden smoothness into singular situations. The idea is that a singular quotient of a smooth variety by a group action retains some of the good properties that the smooth variety has. Stacks are a solid way to formalize this simple-minded idea. We will start with this simple-minded approach, but soon will have to replace it by a more abstract categorical approach. The reason is that studying stacks as “locally quotients of things” is a simple idea but quite tricky to formalize without that other approach. What does it mean to have a map .X → Y of such things? This question is especially thorny when X maps to the fixed-point set of the group action in Y .

D. Abramovich () Department of Mathematics, Brown University, Providence, RI, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_2

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We will therefore turn to the other source of stacks—moduli spaces—and introduce in turn: categories fibered in groupoids; stacks, and then algebraic stacks. This owes a whole lot to Fantechi’s treatment [12], whose title gave birth to the present one. Following the quote above, we will make sure to tether the abstract categorical formalism to the ground level of VARIETIES, specifically those stacks that locally look like quotients of smooth varieties.

1.2

Prerequisites, Readings, Exercises

This text presupposes familiarity with varieties and schemes; sheaves of modules; separated, proper, projective morphisms and blowing up; differentials; flat, étale and smooth morphisms. This and more is covered in Hartshorne’s book [14].

1.2.1 Recommended Reading The following texts take a similarly light point of view on introducing stacks: [1, Lecture 2], [11, 12]. 1.2.2 Further Reading The following texts provide a thorough treatment: [10, Appendix], [21, Appendix], [6, 17, 18, 20]. The text [6] is said to be especially accessible to the uninitiated reader. 1.2.3 Exercises Exercises in this document are an essential component. Some are direct application of definition, but some are indications of major results not covered here.

2

Varieties and Stacks

2.1

What Is a Variety?

So what is a variety, really? We will work with varieties concretely, so much so that, taking a lesson from Fruhbis-Krüger’s lectures, we will be able to tell a dumb computer what a variety is. From this point of view, a variety is a bunch of rings with ring homomorphisms between them, satisfying certain conditions.

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Let us expand this view. We know what an affine variety is, and a variety, after all, is something that locally looks like an affine variety. This implies that a variety X has an open covering .X = ∪Ui with .Ui = Spec Ai affine. But to glue X together from the .Ui a bit more data are needed: we can identify .Uij = Ui ∩ Uj as open subsets of .Ui and .Uj , and there are obvious isomorphisms .φj i : Uij → Uj i . These are not arbitrary—on .Uij k = Ui ∩ Uj ∩ Uk we must have .φkj ◦ φj i ◦ φik = id the identity.  It is common to formalize this as follows: write .V = Ui . Then .V = Spec Ai is affine. The fibered product .R = V ×X V is also affine, since a variety is separated. Write .R = Spec B. As a fibered product R admits two maps t

R ⇒ V.

.

s

This encodes how .Uij sit in .Ui and .Uj , but also the trivial—but indispensable—fact that .Uii → Ui is an isomorphism—with inverse given by the diagonal map .e : V → R, and the fact that .Uij and .Uj i are isomorphic via the reflection .ι : R → R. The triple intersection identity is given in terms of the composition map m : R × R → R.

.

tV s

Such a structure is called a groupoid of schemes. All in all we have two rings .A, B, with two morphisms .s ∗ , t ∗ : A → B, and some other maps we usually neglect to mention: ∗ ∗ ∗ .e : B → A, ι : B → B and .m : B → B ⊗A B, and all these must satisfy a collection of axioms which are obviously true when R and V are given from a covering of a variety X and in general guarantee the existence of such a variety. So indeed a variety is just a bunch of rings with ring homomorphisms between them. This is something a dumb computer is happy to process. Exercise 2.1.1 Construct the affine schemes V and R, with the morphisms between them, giving rise to .P2 from its standard open charts. Indicate the rings and ring homomorphisms.

2.2

What Is a Stack?

In contrast, a stack is a bunch of rings with ring homomorphisms between them, satisfying certain conditions.

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Whait, what? The same thing? There must be a difference, but on this level—of data to give a dumb computer—it is basically the same. Consider the case of a variety V with a finite group-scheme G acting. We will construct, in due time, an algebraic stack .X = [V /G] representing the quotient in a better way than the quotient variety .V /G—whether or not that exists. For now let us see that .X must be a bunch of rings with homomorphisms between them, at least when .V = Spec A is affine and G is a constant group.  Write .R = G × V . Then .R = Spec A|G| = Spec g∈G A is also affine. We have an action .a : R = G × V → V and a projection .pV : R = G × V → V , giving a diagram t=pV

R ⇒ V.

.

s=a

The identity element of G gives .e : V → R, the inverse on G gives .ι : R → R, and the multiplication law on the group gives .m : G × G × V = R ×V R → R. We will see later that when we set these up correctly this makes a groupoid of schemes. It is encoded by the rings .A, B := A|G| , and the homomorphisms .s ∗ , t ∗ , e∗ , ι∗ and .m∗ , which our dumb computer is still happy to process.

2.3

What Is the Difference?

When we describe a variety, s,t

(1) the map .R → V × V is a closed embedding, and (2) the maps .Uij → Ui , Uj are open embeddings, implying that for any .w ∈ R with .s(w) = v ∈ V the homomorphisms of local rings .OV ,v → OR,w is an isomorphism, and similarly for s replaced by t. In contrast we have the following example: Exercise 2.3.1 (1) Consider the action of .G = Z/2Z on the affine line .V = Spec C[x] sending x to .−x. Show that the map .R → V × V is not an embedding. (2) Show that it is nevertheless finite. In order to work with separated stacks, we instead require the map .R → V × V to be finite. In general we also cannot insist on open embeddings, and more seriously, not even an isomorphism of local rings. What we impose instead—for Deligne–Mumford stacks—is

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that the maps .s, t : R → V should be étale meaning that the homomorphisms .s ∗ , t ∗ of sh are isomorphisms. strictly henselian local rings .OVsh,v → OR,w Exercise 2.3.2 (1) Consider the trivial action of the .Q-group scheme G = μn := Spec Q[t]/(t n − 1)

.

on the point .V = Spec Q. Show that the map .R → V × V is not an an isomorphism on local rings whenever .n > 2. (2) Considering the factorization of .t n − 1 to its irreducible components, show that the map .R → V × V is nevertheless étale.

3

Categories for Moduli Spaces

3.1

Moduli

To really understand what stacks are about we change course. Stacks really come about in order to understand moduli spaces, when the moduli problem is not represented by a scheme. We will have two key examples: Example 3.1.1 The moduli stack .Mg of curves of fixed genus .g > 1, and Example 3.1.2 the quotient .[V /G] of a variety V by a finite group G. Special cases of Example 3.1.2 are (1) the quotient .BG := [Spec k/G] of a point by the trivial action of G, and (2) The quotient .V = [V /{1}] of any variety by the trivial group.

3.2

Moduli of Curves as a Category

We consider Example 3.1.1. When we talk about moduli of curves, we want to classify curves of genus g up to isomorphisms. For our discussion, we work over a field k and fix an integer .g > 1, and a curve of genus g is a smooth projective geometrically integral curve X of genus g over some field extension of k. A family of curves of genus g is a projective flat morphism .X → S whose geometric fibers are curves of genus g.

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The great observation of Deligne and Mumford is that the moduli space .Mg is the category of families of curves of genus g. All that one needs to do is to make this category “geometric”. How do we make families of curves of genus g into a category? Say .X1 → S1 and .X2 → S2 are families of curves of genus g. A morphism between them, for the purpose of classification, is a cartesian diagram of schemes

.

Note that this implies that the morphism .X1 → S1 ×S2 X2 is an isomorphism, whose datum is equivalent to the datum of .X1 → X2 by the universal property of fibered products. For the same reason note also that given .X2 → S2 , a family of curves of genus g, and given a morphism .S1 → S2 , the cartesian product .X1 = S1 ×S2 X2 sits in such a diagram, in a way which is unique up to unique isomorphisms. That’s by the universal property of fibered products. Given a scheme S, the fiber of .Mg over the identity morphism of S, denoted .Mg (S), is the subcategory of families .X → S where the morphisms fit over the identity of S:

.

The definition implies that .X1 → X2 is necessarily an isomorphism. This fits with the idea that we set out to classify curves up to isomorphisms.

3.3

A Variety Is a Category

We consider Example 3.1.2(2). Stacks come to extend the category .Sch of schemes. On the other hand, stacks come to encode the idea of a moduli space. In what way is a scheme, or a variety V , a moduli space? The answer is that any variety is the moduli space of its own points. We know what a point on V is. What is a flat family of points? For a scheme S define a family of points on V parametrized by S to simply be a closed subscheme .S ⊂ S × V mapping isomorphically to S. Look at the picture (Fig. 1)—this is the right notion!

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Fig. 1 The graph of a morphism .f : S → V as a family of points on V parametrized by S

Wait—in which way is this a category? If we have .S1 ⊂ S1 × V and .S2 ⊂ S2 × V then we have a morphism precisely when .S1 = S1 ×S2 S2 ⊂ S1 × V . Here there is no choice for a map since we are equating .S1 × V = S1 ×S2 S2 × V . This is equivalent to giving a section .S → S × V , which in turn is equivalent to giving a morphism .S → V . In other words, the moduli category of points on V is the category .SchV of schemes over V .

Exercise 3.3.1 Describe, as a category, the fiber .SchV (S) of the category .SchV over the identity of a scheme S.

3.4

Categories Fibered in Groupoids

We can now generalize: Definition 3.4.1 A functor .F : C → Sch from a category .C to the category of schemes makes .C a category fibered in groupoids if for every object .X2 ∈ C(S2 ) and any morphism of schemes .f : S1 → S2 there is a morphism .f˜ : X1 → X2 such that .F (f˜) = f , which is unique up to unique isomorphism. Remark 3.4.2 This is not my fault—the notion of groupoid in a category fibered in groupoids is not the same as the notion of groupoid in a groupoid in schemes. They are close enough to cause likely confusion. Please be careful! Exercise 3.4.3 Verify that .Mg is a category fibered in groupoids. (You should use the functor that takes a family of curves .C → S to its base scheme S.) Exercise 3.4.4 Verify that .SchV is a category fibered in groupoids. (You should use the functor that takes a “family of points” .S → V to its base scheme S.)

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Recall that a set Z gives rise to a category whose objects are the elements of Z and arrows are declared to be just the ones needed, namely .idz for all .z ∈ Z. Also a category is said to be a set if it is equivalent to a category associated to a set (so it is small and all arrows are unique isomorphisms). Exercise 3.4.5 Show that the fibers of .SchV are sets. We say that a category fibered in groupoids is fibered in sets if the fibers are sets. The exercise shows that the category of points on a scheme is fibered in sets.1

3.5

Arrows

Let .F1 : C1 → Sch and .F2 : C2 → Sch be categories fibered in groupoids. A morphism or base-preserving functor is a functor .G : C1 → C2 with .F2 ◦ G = F1 . Exercise 3.5.1 Show that a morphism .SchS1 → SchS2 is equivalent to a morphism .S1 → S2 . Show also it is equivalent to an object of .SchS2 (S1 ) and to an element of .S2 (S1 ). It thus makes sense to identify S with .SchS . For instance, suppose you have a morphism .ξ : SchS → C. Consider the final object idS idS ¯ of .C(S), and for any other object .S → S of .SchS . Its image .ξ(S → S) is an object .ξ ¯ by g. Conversely, given .g : T → S of .SchS its image is necessarily the pullback of .ξ an object .ξ¯ of .C(S) we obtain a morphism .ξ : SchS → C, assigning to .g : T → S the pullback of .ξ¯ by g. Exercise 3.5.2 Draw a big diagram on a big page (or board, or flip chart) explaining this sentence. It now makes sense to identify: morphisms .S → C, in other words morphisms .SchS → C, with objects .C(S).

3.6

Quotients of Free Actions

Consider now a scheme X with a free action of a group-scheme G having a geometric quotient .X → Y . This precisely means .X → Y is surjective and that G acts simply transitively on geometric fibers of .X → Y —we say that .X → Y is a G-torsor or principal G-bundle.

1 It turns out these are not the only ones.

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We want to say that Y is the moduli space of orbits on X, which precisely means that its category .SchY is naturally equivalent to the “category of families of orbits”, which we must now define. First note: since the action is free, an orbit in X is a principal G-variety P and an equivariant map .P → X. This allows us to define, for such a free action, a “family of orbits” over a scheme S to be a principal G-bundle .P → S with an equivariant map .P → X, a diagram like this

.

A morphism from .(P1 → S1 , P1 → X) to .(P2 → S2 , P2 → X) is defined as a cartesian diagram:

.

Note that whenever we have a morphism .S → Y we may form the principal bundle P := S ×Y X, which forms a diagram as above:

.

.

Conversely, given a principal bundle .P → S and equivariant .P → X, the composite map .P → Y is G-invariant. Now S is the categorical quotient of the action of G on P —this follows for instance by flat descent—so the morphism .P → Y factors uniquely through a morphism .S → Y . This implies that .P = S ×Y X. It follows that the category of families of orbits on X is equivalent to .SchY .

Exercise 3.6.1 Complete a proof of this statement.

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3.7

Quotients in General

We come to Example 3.1.2 in general. Now let G act on X, not necessarily freely. We define a category fibered in groupoids .[X/G] whose objects are diagrams like this

.

and whose arrows from .(P1 → S1 , P1 → X) to .(P2 → S2 , P2 → X) are cartesian diagrams

.

Exercise 3.7.1 (1) Verify that this is a category fibered in groupoids. (2) Verify that .[X/G] is equivalent to .SchY if and only if G acts freely on X having quotient scheme Y . To introduce the following exercises, we need a definition that will serve us well later: Definition 3.7.2 Given categories fibered in groupoids .X, S, and Y , and morphisms .ψX : X → Y, ψS : S → Y , namely morphisms of categories compatible with the functors to .Sch, we define the fibered product .S ×Y X to be the category whose objects over a scheme T are (1) morphisms .T → S and .T → X, equivalently objects .ξS ∈ S(T ) and .ξX ∈ X(T ), and (2) an isomorphism .ψS (ξS ) → ψX (ξX ). Exercise 3.7.3 Write explicitly what this means when .S, X are schemes. Exercise 3.7.4 Check that this is compatible with pullbacks, making .S ×Y X a category fibered in groupoids.

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Exercise 3.7.5 Going back to where .X, S are schemes and .Y = [X/G], check that the resulting diagram

.

is cartesian. In other words, categories fibered in groupoids allow us to have true quotients in general, extending the notion for free actions.

Example 3.7.6 Consider now the trivial action of a group-scheme on .Spec k. It is customary to write .BG := [Spec k/G]—it is known as the classifying stack of G, at least once it receives the structure of a stack. Objects over a k-scheme S are principal G-bundles .P → S, and arrows are cartesian squares. Exercise 3.7.7 Given a G-equivariant morphism .X1 → X2 construct a natural morphism [X1 /G] → [X2 /G].

.

This in particular provides a morphism .[X1 /G] → BG for any k-scheme X. Exercise 3.7.8 Given a homomorphism .H → G and an action of G on X construct a natural morphism .[X1 /H ] → [X2 /G]. The next sections investigate how close we can get categories fibered in groupoids to being “geometric”. Quotients, specifically by tori, are a central object of [22] in this volume.

4

Stacks

The category .Sch of schemes has some useful Grothendieck topologies—in particular there is a sense in which a morphism .Si → S is étale or smooth; and if this morphism is also surjective then it is an étale cover, or a smooth cover (namely a cover in the smooth topology); if .Si are Zariski open subsets you get, of course, a Zariski cover. To avoid saying “Zariski or étale or smooth” all the time we might refer to a .T -covering, with .T indicating the chosen topology.

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4.1

Descent for Morphisms of Curves

Let .C1 → S and .C2 → S be two families of curves of genus g. What would it take to show that they are isomorphic? Of course this requires writing an S-morphism .C1 → C2 and an inverse, but to do it concretely—say to give a dumb computer a morphism—we need to use a covering. After all morphisms are locally defined. Say we have a Zariski covering .Si → S and an isomorphism .φi : C1i → C2i of the pullbacks of .C1 , C2 to .Si , then by definition of a morphism this gives an isomorphism .φ : C1 → C2 if and only if .φi agree on the intersections: .φi |Sij = φj |Sij . It is not trivial, but not hard either, that the same is true for the étale or smooth topologies: if .Si → S is a .T -covering then a collection of isomorphisms .φi : C1i → C2i comes from a morphism .φ : C1 → C2 if and only if .φi agree on the “intersections”: .φi |Sij = φj |Sij , where .Sij = Si ×S Sj . It is customary to assign to .C1 → S and .C2 → S the functor .IsomS (C1 , C2 ) : SchS → Sets. For an S-scheme T its value is the set .IsomT (C1T , C2T ) of isomorphisms of .C1T → C2T . The discussion above says that this functor is a .T -sheaf : for a .T -covering .Ti → T we have an equalizer sequence—the analogue of an exact sequence of sets: .

Exercise 4.1.1 Consider the case where .Ci → S is a family of curves of genus 0. What does the sequence above say about comparing families of .P1 ’s over a base S?

4.2

Descent for Curves

Let S be a scheme. What would it take to construct a family of curves .C → S? Again one needs to work locally. But now if we are given families of curves .Ci → Si , their gluing requires the additional data of isomorphisms .φj i : Ci |Sij → Cj |Sij . And such data .φij must satisfy the proverbial cocycle condition on .Sij k . One can interpret this in terms of a longer, and categorical, “exact sequence”, but let’s leave that to the .∞-categorists. The traditional, 1-category language is to say that “curves satisfy effective descent”, namely the families .Ci → Si and isomorphisms .φj i give rise to .C → S, unique up to a unique isomorphism, if and only if the cocycle condition holds. This holds for the Zariski topology by the definition of a scheme (and set-theoretic gluing). Again it is not trivial, but not hard either, that the same is true for the étale or smooth topologies. Exercise 4.2.1 Consider the case where .Ci → S is a family of curves of genus 0. What does this discussion say about comparing families of .P1 ’s over a base S, in the Zariski or étale topology?

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75

Stacks in General

Let us now fix .T = Zariski, étale or smooth. A category fibered in groupoids .C is a stack if Isom functors are .T -sheaves and if every descent datum is effective. The discussion above says that .Mg is a stack in any of these topologies. It is not too hard to show that the same is true for the categories .[V /G] we discussed earlier. In particular the category .SchV is a stack. Exercise 4.3.1 Outline for yourself why indeed .Mg , SchV , and .[V /G] are stacks in the étale topology.

4.4

Discussion

To summarize, stacks allow us to put meaningful topological structures on categories fibered in groupoids, in particular those associated with natural moduli spaces and with quotients.

This in particular means that objects of our category—or moduli problem—are local in nature. For instance a family .C → S of curves of genus g over S can be recognized as such by restricting it to a covering .Si → S. From a practical point of view, this also means that our proverbial dumb computer can work with it, taking charts .Si on S and families .Ci → Si , with appropriate gluing data on the overlaps. Note however that objects of stacks are one level more complex than sections of sheaves: to give a section .s ∈ F(S) of a sheaf of sets .F on a scheme S covered by .Si , it is equivalent to give sections .si ∈ F(Si ) which agree on the overlap. The first sheaf axiom says that given .si such s is unique if it exists—a separatedness condition. The second sheaf axiom says that if .si agree on overlaps such s does exist—a locality condition. This is all that is needed to glue sets, in essence since a bijective map of sets can be broken down to being injective and surjective. In the case of stacks, we are gluing categories. Keeping the analogy with bijections of sets, here we are concerned with equivalences of categories, a slightly more subtle question. Recall that to verify that a functor .G : A → B is an equivalence of categories, one checks three conditions: two conditions on morphisms—that .G is full (surjective on arrows) and faithful (injective on arrows), and a third condition on objects—that .G is essentially surjective. For stacks, the analogy leads to the two sheaf conditions for arrows, and a third condition on objects, requiring them to glue locally. This is necessarily more subtle—it involves triple intersections and the cocycle condition.

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5

Algebraic Stacks

5.1

Representability

A stack .C is represented by a scheme S if it is isomorphic to .SchS . It is representable if it is represented by some scheme. A morphism .C1 → C2 of stacks is representable if for any scheme S and any morphism .S → C2 the fibered product .S ×C2 C1 is a scheme.2 One can show that if T is a scheme, then a morphism .ξ : T → C is representable if and only if for any scheme S and .η : S → C the sheaf .IsomS×T (ξ, η) is a scheme.3 Exercise 5.1.1 Show that any morphism .S → C is representable if .C = SchV is the stack represented by a scheme V . Exercise 5.1.2 Show that any morphism .S → C is representable if .C = [V /G]. Exercise 5.1.3 Show, using Hilbert schemes, that any morphism .S → C is representable if .C = Mg .

5.2

Smooth and étale Morphisms

A representable morphism .C1 → C2 is said to be smooth if for every scheme S and .S → C2 the resulting morphism of schemes .S ×C2 C1 → S is smooth. Similarly for étale, smooth covering, étale covering, proper, etc.

5.3

Algebraic Stacks

A stack .C is .T -algebraic if (1) for any scheme S, any .S → C is representable; equivalently the diagonal morphism .C → C × C is representable, and (2) there exists a .T -covering .V → C. Generally, étale-algebraic stacks are known as Deligne–Mumford stacks, and smoothalgebraic stacks are known as Artin stacks.

2 The standard terminology allows it to be an algebraic space. Our more restrictive notion will suffice

for our purposes. 3 With the algebraic space terminology this is automatic, since reasonable sheaves are algebraic

spaces.

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Exercise 5.3.1 Show explicitly that when .C = SchV , a morphism .S → C is smooth/étale etc. if and only if .S → V has the same property. Deduce that .SchV is algebraic. Exercise 5.3.2 Show explicitly that when .C = [V /G], a morphism .S → C is smooth/étale etc. if and only if .P → V has the same property, where .P = S ×C V . Show that, if G is smooth, .V → C is a covering in the smooth topology. Deduce that .C is algebraic. M. Artin has shown how to relax the requirement that G be smooth. Exercise 5.3.3 Every smooth curve of genus 0 has an anticanonical embedding in .P2 . Considering the Hilbert scheme of conics in .P2 , show that the locus of smooth conics is open. Show that .M0 is the quotient of this scheme by the action of a smooth group scheme, hence is an algebraic stack. Is it a Deligne–Mumford stack? Exercise 5.3.4 Can you repeat the discussion above for .M1,1 , the moduli stack of elliptic curves? What happens with .M1 , the moduli stack of curves of genus 1? Exercise 5.3.5 This is the crux of Edidin’s paper [11], and you might wish to just read Edidin’s account: (1) Show using Riemann-Roch that the three-canonical linear series separates points and tangent spaces: .dim H 0 (C, OC (3KC )) = dim H 0 (C, OC (3KC − p − q)) + 2 for any two points .p, q. Deduce that every smooth curve of genus g has a complete linear series giving a 3-canonical embedding into a projective space of dimension .5g − 6. (2) Show that such curves form a locally closed subscheme H of the appropriate Hilbert scheme, parametrizing curves of genus g of degree .6g − 6 inside .P5g−6 . (3) Show that the quotient .[H /G] with .G = P GL5g−5 is isomorphic to .Mg , as follows: ⊗3 • Given a family of curves .π : C → S, consider the locally free sheaf .F = π∗ ωC/S and its projective frame bundle .P(F) → S. Show that this is a G-bundle. Show that the pullback of the projectivization .P(F) to .P is a trivial projective bundle, in which the pullback of C embeds as a 3-canonical curve. Show that this gives a G-equivariant morphism .P → H , inducing a morphism .S → [H /G]. In turn this defines a morphism .Mg → [H /G]. • There is a universal curve .CH → H embedded in .H × P5g−6 . Given a principal Gbundle .P → S and equivariant .P → H , consider the pullback .CP of this universal curve. Show that G acts freely on .CP → P, and the quotient is a family of genus g curves .C → S. This in turn defines a morphism .[H /G] → Mg .

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• One needs to show that these morphisms are quasi-inverses of each other (a task rarely seen in writing). (4) Deduce that .Mg is algebraic.

5.4

Smoothness of Quotients

Traditionally one defines a morphism .f : C1 → C2 of algebraic stacks to be smooth if it is of finite presentation and satisfies the infinitesimal criterion for smoothness. A bit of diagram chasing allows one to circumvent this: the infinitesimal criterion holds if and only if, given a smooth covering .V → C1 , the morphism .C1 → C2 is smooth if and only if the composite .V → C2 is smooth. Let us now consider the quotient .[V /G] of a k-variety V by a smooth group-scheme G. we conclude that .[V /G]

→ Spec k is smooth if and only if .V → Spec k is smooth.

In other words, algebraic stacks are indeed a way to find hidden smoothness in quotients!

6

Constructions

6.1

Quotients and Coarse Moduli Spaces

Consider now a scheme X with an action of a group G. A quotient scheme .X → Y = X/G is a G-invariant morphism of schemes such that (1) if .X → Z is another G-invariant morphism of schemes then it factors uniquely as .X → Y → Z, and (2) For any algebraically closed field K, the map .X(K)/G(K) → Y (K) is bijective. The first condition shows that a quotient is unique up to unique isomorphism, and the second says that in a sense points of the quotient are orbits of points on X. Symmetric functions allow you to prove: Exercise 6.1.1 Let .X = Spec A be an affine scheme and G a finite group. Show that the natural morphism .X → X/G := Spec AG is a quotient scheme.

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In general quotients do not exist in the category of schemes. If G is finite then quotients exist in the category of algebraic spaces, but this is a cheat: in a sense it is an outcome of the existence of the quotient stack. Exercise 6.1.2 Let X be a scheme with an action of a group G, and assume a quotient scheme .X/G exists. Show that .X → X/G factors uniquely as .X → [X/G] → X/G. So in a sense .X/G is the best schematic approximation (from below) of the stack [X/G]. What can we do in general? Consider a stack .X . In analogy to the definition of a quotient, Mumford defined the following:

.

Definition 6.1.3 A coarse moduli space .X → X is a morphism to a scheme or algebraic space X such that (1) if .X → Z is another morphism to a scheme then it factors uniquely as .X → X → Z, and (2) For any algebraically closed field K, the map .X (K) → X(K) is bijective. This is quite relevant to our volume, especially the following: Exercise 6.1.4 Let X be a scheme with an action of a group G, and assume a quotient scheme .X/G exists. Show that .[X/G] → X/G is a coarse moduli space. There is a general result. One defines a stack to be separated if the same definition as for schemes applies. Keel and Mori [16] proved the following fundamental result: Theorem 6.1.5 Let .X be a separated Deligne–Mumford stack. Then it admits a separated coarse moduli space .X → X. This applies to .Mg , g ≥ 2 and to .[X/G], with G a finite group. It means that any separated Deligne–Mumford stack can be well-approximated (from below) by an algebraic space.

6.2

Root Constructions

Suppose S is a scheme, and .f ∈ Γ (OS ) a function. A classical construction is the n-th order cyclic cover .S → S along the divisor .D = V (f ) given by the equation .y n = f . In other words .T = Spec OS [y]/(y n − f ). This is a very useful construction in geometry, but there is something unsatisfying about it: it really does depend on f , not only on .V (f ): Say you replace f by .f = uf , where u

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is unit. Unless u happens to be an n-th power, the s-schemes .T = Spec OS [y]/(y n − f ) and .T = Spec OS [y]/(y n − f ) are simply not isomorphic. This means that one needs to be careful to globalize the construction. Classically one globalizes the situation by choosing a line bundle L and an isomorphism n  O(D), which one rather writes as .φ : L−n → I ⊂ O . In this case T is the .L D D  quotient of the algebra .R = n≥0 L−n by the equations .s n − fs where s is a local section of .L−1 and .fs = φ(s n ) is the corresponding local function vanishing on D. Exercise 6.2.1 By choosing a generating section of L, show that locally this construction coincides with the scheme T constructed above. What happens if one replaces .φ by .uφ for a unit u? Even putting units aside this still depends on a choice—replacing L by another n-th root of .O(D) changes the cover. Even worse—an n-th root L might not exist globally. The Cadman–Vistoli root stack comes to fix these issues. Exercise 6.2.2 (1) Consider the category whose objects over S are line bundles on S and arrows are pullback diagrams. Show that it is equivalent to .BGm , in particular an Artin stack. Show that the functor sending L to .Ln corresponds to the n-th power map .Gm → Gm . (2) Now consider the category whose objects over S are pairs .(L, s) consisting of a line bundle L and a section s. Show that it is equivalent to .A1 := [A1 /Gm ] with the natural action. Show that the functor .A1 → A1 sending .(L, s) to .(Ln , s n ) corresponds to teh n-th power map on .A1 → A1 , with the corresponding n-th power map on .Gm . Exercise 6.2.3 Consider a scheme V , and a line bundle L on V with section s—for the √ discussion let us assume giving a Cartier divisor D—and a positive integer n. Let . n (V , D) be the category whose objects over S are triples .(f, M, φ, t) where .f : S → V is a morphism, M an invertible sheaf on S with section t, and .φ : M n → f ∗ L an isomorphism carrying .t n to s. Show that this is an algebraic stack, isomorphic to .V ×A1 A1 , where the map on the right is the n-th power map. Exercise 6.2.4 Suppose D is the vanishing locus of a function g. Let T be the scheme defined at the beginning of the section. Show that .(f, M, φ, t) is isomorphic to the quotient .[T /μn ], where .μn acts on y by multiplication, in particular it does not depend on the choices discussed above.

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81

Coherent Sheaves

This is definitely a topic that deserves thorough attention, but not in a light-touch document such as this. The ideas are natural and mostly extend what you know about varieties smoothly. As Deligne–Mumford stacks admit étale covers by schemes, they have a well defined élate topology. This in particular means that one can consider sheaves in the étale topology of a Deligne–Mumford stack. One simple example is the structure sheaf .OX of a stack .X , represented by the structure sheaf on an étale covering .V → X , with trivial gluing data on the overlaps. This manifests again the observation that a stack is just a bunch of rings with rings homomorphisms between them. . . One can similarly consider sheaves of .OX -modules, quasicoherent and coherent sheaves of .OX -modules. Ideals, differential forms, and other related notions, such as the spectrum and projective spectrum of a sheaf of algebras, are studied as for schemes. The situation is not quite as simple for Artin stacks, but this difficulty is better swept under the rug in the present exposition.

6.4

Stack Theoretic P roj

Let V be a scheme and .A = ⊕m≥0 Am a finitely generated quasi-coherent .OV -algebra, with its natural ideal .A+ = ⊕m≥1 Am . Consider .SpecS A with its vertex .V+ = V (A+ ). The grading gives a .Gm -action. Define ProjV (A) := [(SpecV (A) V+ ) / Gm ]. Exercise 6.4.1 Consider the special case where .A is generated by .A1 as an .A0 -algebra. Show that in this case .ProjV (A) = ProjV (A), the standard relative projective scheme construction found, for instance, in Hartshorne’s book.

6.5

Weighted Projective Stacks

The .Proj construction gives something new already when .V = Spec k is a point. Consider the algebra .A = k[x1 , . . . , xn ], but with the variables .xi placed in degree .wi > 0. In other words, the multiplicative group .Gm acts on .xi via .xi → t wi xi . Note that .Spec A is just affine space, and .V (A+ ) is the origin. The quotient stack ProjV (A) := [(SpecV (A) V+ ) / Gm ] is the weighted projective stack .P(w1 , . . . , wn ) of dimension .n − 1. Its coarse moduli space (SpecV (A) V+ ) / Gm is the classical weighted projective space .P(w1 , . . . , wn ).

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Blowups and Weighted Blowups

Consider now the situation where .A = ⊕I m , where .I is a sheaf of ideals. Then .Proj A = Proj A is the blowup of the subscheme .V (I) in the classical case. When V and .X = V (I) are smooth, namely .I is everywhere generated by a partial local system of parameters .x1 , . . . , xk on the smooth variety V , we obtain the familiar smooth blowup of X on V . A somewhat more general stack theoretic construction is associated with Rees algebras. Say the graded pieces of .A = ⊕Im are nested ideals sheaves with .Im ⊃ Im+1 , and multiplication is given by multiplication of ideals, in particular .Im In ⊂ Im+n . This is the stack theoretic blowup of the Rees algebra .A. Finite generation implies that its coarse moduli space coincides with the blowup of .Im for some large and divisible m, but the stack has richer structure. A central object of this book are smooth weighted blowups. It is discussed in several sections with different emphases, different levels of generality, and different points of view. We describe here only the local case, where the center is .V (x1 , . . . , xk ), with .xi a partial regular system of parameters, and .xi given weight .ai . This is obtained by the Rees algebra .A = ⊕Im described as follows: one starts with the algebra .B over .OV where .xi is placed in .Iwi . This is already an algebra of ideals, but to make it a Rees algebra one needs to enforce the condition .Im ⊃ Im+1 , in essence by  replacing .Bm by the ideal .Im = j ≥m Bm . More natural presentations, involving valuations or extended Rees algebras, are presented in the following chapters.

6.7

Destackification

With the exception of the classical method described in [13] in this volume, there is one point in all the recent work which we have found challenging to explain, and I wish to try to dispell this challenge. The point is that all the new methods start with a scheme X and end up with a smooth stack .X resolving it. Evidently algebraic geometers want resolution of singularities to end up with a scheme, no matter how useful stacks may be. Have we ended up short of this goal? Our answer is, and always has been, that the smooth stack .X always admits a destackification, which is an easily understood and computationally feasible task. This task works even in positive characteristics, as long as the stack .X is tame, a condition which is automatic in characteristic 0. It applies in pure or mixed characteristic for all modification methods discussed in this volume. What is destackification? Definition 6.7.1 Let .X be a smooth Deligne–Mumford stack. A destackification of .X is a proper birational morphism .X

→ X from another stack, such that the coarse moduli space .X

of .X

is smooth.

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The universal property of coarse moduli spaces implies that, if .X is the coarse moduli space of .X , there is a proper birational morphism .X

→ X induced by .X

→ X . In particular .X

→ X → X is a resolution of singularities in the classical sense. One can envision stronger statements. For instance one may wish the stack .X

to also be smooth, and one may wish it to be obtained by a sequence of simple operations like smooth blowups and smooth root constructions. One may further wish .X

→ X

to be very simple—a sequence of smooth root constructions. All these do hold true, and were developed in different works. For the present discussion all we need is the existence of a destackification, and its inherent simplicity, in particular it is much simpler than any general algorithm of resolution of singularities. For the sake of discussion, let us note that all the stacks .X appearing in our work have at most finite abelian (and tame) stabilizers, acting faithfully on tangent spaces. One can say a whole lot in greater generality, but let us stick with this situation for simplicity. There are several ways to achieve destackification.

6.7.1 Direct Resolution Over the years people have devised, again and again, methods for resolving varieties with finite abelian tame quotient singularities, such as the space .X . An early work in this direction is due to Bogomolov [9, Lemma 8.2], which addresses global quotients. Such resolution .X

→ X is all we need, but in fact it does provide a destackification, with possibly singular .X

, if one simply takes .X

= X

×X X , whose coarse moduli space is

.X . 6.7.2 Torification and Toroidal Resolution When passing to strict henselizations, the action of the stabilizers of .X on the tangent spaces can be diagonalized, but this diagonalization is not canonical and cannot be glued to a global structure. However, the works [2–4] provide the following result: Theorem 6.7.2 The stack .X admits a canonically defined ideal sheaf .ITor whose blowing

→ X , endowed with its exceptional divisor, is toroidal, and the stabilizers on .X

up .XTor Tor are globally diagonalized with respect to this divisor. The ideal .ITor , the so-called torific ideal, is obtained locally as the product of ideals of the form .Iχ , where .χ runs over the characters of the local stabilizer .Gx : Iχ

.

=

  sh f ∈ OX,x : g ∗ f = χ (g) · f ∀g ∈ Gx .

This implies that the coarse moduli space .XTor is toroidal, and its resolution of





singularities .X → XTor , with pullback stack .X = X

×X X , provides a very simple destackification as above. A similar procedure was devised by Gabber, see [15].

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6.7.3 Strong Destackification The works [7, 8] provide the strongest destackification result, with all steps being simple operation. It is, however, significantly more costly in computational terms: Theorem 6.7.3 There exists a sequence of proper birational maps .X

= Xn → · · · → X1 → X0 = X each of which being either a smooth blowup or a root construction along a smooth divisor, and a sequence of root constructions along smooth divisors .X

= Xn

→ · · · → X0

= X

, In particular .X

→ X is a destackification. A computer implementation of this algorithm in OSCAR [19] is underway, see [5].

6.8

Quotients By Groupoids [R ⇒ V ]

Kai Behrend gave an elegant description of the stack associated to a groupoid in general. First we note the following: for any scheme X and any étale surjective .V → X, one can write .RV = V ×X V and the two projections give a groupoid .RV ⇒ V . If, as suggested above, .RV is to be considered as an equivalence relation on V , then clearly the equivalence classes are just points of X, so we had better define things so that .X = [RV ⇒ V ]. Now given a general groupoid .R ⇒ V , an object over a base scheme B is very much like a principal homogeneous space: it consists of an étale covering .U → B, giving rise to .RU ⇒ U as above, together with maps .U → V and .RU → R making the following diagram (and all its implicit siblings) cartesian:

.

There is an important object of .X = [R ⇒ V ] with the scheme V as its base: you take U = R above, with the two maps .U → B and .U → V being the source and target maps .R → V respectively. What it does is it gives an étale covering .V → X , as required in Sect. 5.3. .

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References 1. D. Abramovich, Lectures on Gromov-Witten invariants of orbifolds, in Enumerative Invariants in Algebraic Geometry and String Theory. Lecture Notes in Mathematics, vol. 1947 (Springer, Berlin, 2008), pp. 1–48 2. D. Abramovich, A.J. de Jong, Smoothness, semistability, and toroidal geometry. J. Algebraic Geom. 6(4), 789–801 (1997) 3. D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk, Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002) (electronic) 4. D. Abramovich, M. Temkin, J.A. Włodarczyk, Toroidal orbifolds, destackification, and Kummer blowings up. Algebra Number Theory 14(8), 2001–2035 (2020). With an appendix by David Rydh 5. D. Abramovich, N. Ben Abla, D. Bhatia, L. Kästner, C. Li, D. Li, D. Lister, S. Obinna, D. Silverston, H. Talbott, J. Wong, An Oscar Implementation of Algorithms A-D of Bergh’s Destackification (GITHUB library, 2021) 6. J. Alper, Notes on stacks and moduli. https://sites.math.washington.edu/~jarod/moduli-7-1222.pdf. Working draft, 12 Jul 2022 7. D. Bergh, Functorial destackification of tame stacks with abelian stabilisers. Comp. Math. 153(6), 1257–1315 (2017) 8. D. Bergh, D. Rydh, Functorial destackification and weak factorization of orbifolds (2019). arXiv e-prints. arXiv:1905.00872 9. F.A. Bogomolov, Stable cohomology of groups and algebraic varieties. Mat. Sb. 183(5), 3–28 (1992) 10. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36, 75–109 (1969) 11. D. Edidin, Notes on the construction of the moduli space of curves, in Recent Progress in Intersection Theory (Bologna, 1997). Trends in Mathematics (Birkhäuser Boston, Boston, 2000), pp. 85–113 12. B. Fantechi, Stacks for everybody, in European Congress of Mathematics, Vol. I (Barcelona, 2000). Progress in Mathematics, vol. 201 (Birkhäuser, Basel, 2001), pp. 349–359 13. A. Fruhbis-Krüger, A Computational View on Hironaka’s Resolution of Singularities (2023, this volume) 14. R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, vol. 52 (Springer-Verlag, New York, 1977). 15. L. Illusie, M. Temkin, Exposé VIII. Gabber’s modification theorem (absolute case). Astérisque 363–364, 103–160 (2014). Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. 16. S. Keel, S. Mori, Quotients by groupoids. Ann. Math. (2) 145(1), 193–213 (1997) 17. G. Laumon, L. Moret-Bailly, Champs Algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 39. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] (Springer-Verlag, Berlin, 2000) 18. M. Olsson, Algebraic Spaces and Stacks. American Mathematical Society Colloquium Publications, vol. 62 (American Mathematical Society, Providence, 2016) 19. OSCAR Development Team, OSCAR—Open Source Computer Algebra Research system (2022). https://oscar.computeralgebra.de

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20. Stacks Project Authors, Stacks Project. http://stacks.math.columbia.edu 21. A. Vistoli, Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math. 97(3), 613–670 (1989) 22. J. Włodarczyk, Weighted resolution of singularities—a Rees algebra approach (2023, this volume)

Introduction to Logarithmic Geometry Michael Temkin

1

Introduction

These chapter is based on a minicourse on logarithmic resolution of singularities given by the author, and it provides an extended version of its first part devoted to introduction to logarithmic geometry with a view towards applications to resolution. I do not aim to build a theory with proofs (and this is impossible in a 3–4 lecture long course). The goal is to make the reader familiar with basic definitions, constructions, techniques and results of logarithmic geometry. I formulate most of the results as “Exercises” and try to keep them at a reasonable level of difficulty. References to the literature are also provided. At the first reading of the material it may be worth just to read the formulations and hints or comments about main ideas of the arguments, without trying to solve them or read proofs in the cited papers.

This research is supported by BSF grants 2014365 and 2018193, ERC Consolidator Grant 770922— BirNonArchGeom. M. Temkin () Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_3

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History and Motivation

1.1.1 The Discovery Logarithmic structures and schemes were discovered by J.-M. Fontaine and L. Illusie on Sunday, July 17, 1988 during a discussion in a train on their travel to Oberwolfach workshop “Aritmetische Algebraische Geometrie”. In fact, the discussion was in the continuation of an IHES seminar that had taken place in the spring, and the construction was motivated by the necessity of finding a suitable framework in which an analogue of Steenbrink’s limiting Hodge structure for a semistable reduction over a complex disc could be defined in mixed characteristic in order to make sense of the .Cst -conjecture of Fontaine and Jannsen. During the workshop Illusie prepared a short summary of the discussion and showed it to K. Kato, who was very enthusiastic about the new notion and very quickly wrote the first paper, where these notions were introduced: “Logarithmic structures of Fontaine-Illusie”. The new theory turned out to be extremely useful because of the following features: (1) It provides a more general notion of smoothness, which allows to work with many classical non-smooth objects similarly to the smooth ones. In particular, it conceptually adjusts various cohomology theories to this generalized context. (2) It conceptually treats various notions of boundaries, such as normal crossings divisors, and it often provides a functorial way to compactify various moduli spaces—smooth objects often degenerate to logarithmically smooth (but non-smooth) objects over the boundary. (3) It provides a conceptual way to bookkeep information on closed subspaces and fibers, in particular, leading to better solutions of deformation problems. Off course these three large classes of properties are tightly connected and often show up altogether.

1.1.2 Precursors In fact, log geometry had numerous precursors, which it absorbed and generalized. Without pretending to provide a full list, here are a few most important ones, which will be discussed in Sect. 2 in more detail: (1) Normal crossings divisors, especially, when viewed as a boundary used to compactify a smooth variety, correspond to log structures. In fact, a smooth variety with a normal crossing divisor is nothing else but a log smooth variety, which is smooth. (2) Toroidal geometry, which was introduced in [7] to prove semistable reduction theorem, is, in fact, the theory of log smooth log varieties. Toroidal morphisms between toroidal varieties are nothing else but log smooth morphisms. (3) Deligne’s generalized divisors and logarithmic structures of Deligne-Faltings, [5, Complement 1]. See also [4, Section 2], where Deligne’s letter and its influence on the discovery of log schemes is described.

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(4) Logarithmic differentials, logarithmic versions of various complexes, etc., which were defined ad hoc, obtain a conceptual interpretation in log geometry. (5) Semistable morphisms are log smooth, so semistable reduction theorem literally becomes a desingularization theorem in log geometry. Furthermore, the snc divisor sitting in the closed fiber of a semistable family is log smooth over the log point—a more exotic object, which book keeps the log structure purely algebraically. In a sense, a log point is a logarithmic analogue of usual non-reduced (or fat) points in the theory of schemes. In fact, the role of log geometry in some other classical problems is being gradually clarified even nowadays. For example, it was clear that it is involved in resolution of singularities, at least through the exceptional divisor, but a careful study of this question not only shed a new light on known methods, but also led to discovery of a new generation of methods, which will be discussed in another chapter. Also, I think that the role of log geometry in compactifying various moduli spaces is not fully exploited yet, and it will increase in future research. This is tangential to the material we want to cover, so we will only discuss a couple of such examples in the sequel and refer the reader to [1], where the theory of log schemes is described from the point of view of applications to the theory of moduli spaces. Our main motivation is application to resolution of singularities in the chapter on logarithmic and relative resolution of singularities.

1.2

Structure of the Chapter

1.2.1 Overview The chapter starts with Sect. 2 on precursors of logarithmic geometry: we discuss the log structure encoded by snc divisors, which is probably the first time log structures implicitly showed up in mathematics, and then recall the more general theory of toroidal varieties. Log schemes are introduced in Sect. 3. We first study necessary properties of monoids and introduce log structures, and in the end of the section we discuss log regular log schemes. Various properties of morphisms: charts, log smoothness and log étaleness, log differentials and log blowings up are reviewed in Sect. 4. Finally, in Sect. 5 we discuss Olsson’s stacks .LogX and the technique of reducing log geometry to geometry of stacks. This is the most technically demanding section, and the only one in which stacks are used. It will be used in the construction of the relative desingularization functor, but not in the more basic case of the absolute logarithmic desingularization.

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1.2.2 References and Sources The single reference with all foundations worked out in detail is the recent book of Arthur Ogus [13]. Originally, logarithmic geometry was established by Kazuya Kato in [5], and the theory of log regular log schemes, their desingularization and log blowings up was developed in [6] and [11]. Stacks .LogX were introduced and their relation to logarithmic properties was studied by Martin Olsson in [14], and some further results were obtained in [8]. 1.2.3 Conventions Often we write “log” instead of “logarithmic”.

2

Precursors

In this section we discuss some situations, where log structures are implicit actors. Later on they will serve as a source of examples and illustrations.

2.1

Normal Crossings Divisors

Definition 2.1 Let X be a regular scheme and D → X a divisor. (i) One says that D is strictly (or simple) normal crossings or just snc at x ∈ X if locally one has that D = V (t1 . . . ts ), where t1 , . . . , tn is a regular family of parameters at x and we use notation V (f1 , . . . , fn ) = SpecX (OX /(f1 , . . . , fn )). (ii) One says that D is normal crossings at x if it is snc étale-locally at x. A divisor is called normal crossings or snc if this is so everywhere on X. Exercise 2.2 (i) Show that the number s = s(x) in the definition is the number of branches of D at s (e.g. the number of irreducible formal components) and hence is an invariant of D at x, also called the multiplicity of D at x. (ii) Define a stratification of D by the multiplicity and show that each stratum D(s) is a regular locally closed subscheme. (iii) Assume that X is regular and D = ∪i∈I Di is a reduced divisor with irreducible components Di . Show that D is snc if and only if for each J ⊆ I the scheme theoretic intersection DJ = ∩j ∈J Dj is regular. Remark 2.3 Local computations with nc (resp. snc) divisors are done using étale (resp. Zariski) neighborhood, where it is given by the vanishing of the product of a subset of a family of regular parameters.

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Toroidal Schemes

2.2.1 Toric Schemes Let M be a lattice and .N = Hom(M, Z) the dual lattice. This can be encoded by the nondegenerate pairing .M × N → Z, but despite the symmetry, we will always view N and .NR = N ⊗ R as geometric spaces, while the elements of M will be viewed as functions .N → Z or .NR → R. Let .σ ∈ NR be an M-rational polyhedral cone, i.e. a cone given by finitely many conditions .mi (x) ≥ 0 with .mi ∈ M. One associates to .σ an affine toric k-variety .Tσ as follows. The dual cone .σ ∨ = {z ∈ MR | z(σ ) ≥ 0} is also rational and it follows easily that the monoid .Mσ := σ ∨ ∩ M is finitely generated. Therefore, .Aσ := k[Mσ ] is an affine k-algebra and .Tσ := Spec(Aσ ) is an affine variety. Note that .Tσ is provided with the natural action of the torus .T0 = DM = Spec(k[M]) whose lattice of characters is M. Furthermore, .Tσ contains an open orbit isomorphic to .T0 , and this is the source of the terminology. The information encoded in the pair .(M, σ ), often called the combinatorial information, is equivalent to the information encoded in .Tσ with the torus action, and there is a very tight and natural relation between the combinatorial and geometric pictures: Exercise 2.4 (i) Show that .(M, σ ) can be reconstructed from the affine toric variety .Tσ as follows: M is the lattice of characters of .T0 and giving the .T0 -action on .Spec(A) is equivalent to providing an M-grading of A, where m-homogeneous elements are the equivariant ones with the character m: they are acted on via the rule .t (f ) = m(t)f ; the cone .σ is determined by the monoid .Mσ , which is precisely the set of characters .m ∈ M with a non-zero homogeneous component .(Aσ )m . (ii) Show that the set of orbits of .Tσ is in a natural one-to-one correspondence with the faces of .σ . In particular, the open orbit .T0 corresponds to the vertex 0 of .σ . Remark 2.5 The affine theory can be globalized as follows. On the schematic part of the picture one glues affine toric varieties .Ti with the same torus .DM along isomorphic open affine toric subvarieties. A resulting object is called a toric scheme with respect to the torus .DM . On the combinatorial side one glues polyhedral cones .σi ⊂ NR along faces. The resulting objects are called cone complexes. Sometimes one only considers complexes embedded in .NR and calls them polyhedral fans. If the union of all faces of a fan is the whole .NR , the fan is called a subdivision. In fact, any separated toric variety corresponds to a fan and a proper toric variety corresponds to a subdivision. We do not go into details and recommend a standard literature, e.g. [3].

2.2.2 Monoidal Resolution of Singularities In the geometry of polyhedral complexes regular simplicial cones play the role of nonsingular points.

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Definition/Exercise 2.6 (i) A polyhedral cone .σ is called regular if its sharpening .M σ = Mσ /(Mσ )× is a free Nr ⊕ Zs . Show that this happens if and only if there monoid .Nr and hence .Mσ → exists a basis .e1 , . . . , en of N such that each edge of .σ contains some .ei . Thus, .σ is a simplicial cone of a very special form. (ii) A polyhedral complex is called regular if all its cones are. A combinatorial or monoidal resolution of singularities is the following result: Theorem 2.7 Any polyhedral complex possesses a regular subdivision. Remark 2.8 We will not prove this theorem, but only make a couple of remarks. One can construct such subdivision by applying functorial (hence equivariant) resolution of singularities to toric varieties and translating it to the combinatorial language. Clearly, this is a too complicated solution that does not admit a simple combinatorial interpretation. A construction of a simple canonical solution of this problem was missing in the literature until very recently, see [15, Theorem 4.6.1], but various non-canonical solutions were well known. First, the claim easily reduces to the case of a single cone. Second, using the barycentric subdivision one reduces to the case of a simplicial cone. Then one defines some invariants of the singularity (essentially, they measure the discrepancy between .σ ∩N and the monoid generated by the elements of N lying on the edges) and finds simplicial subdivisions that decrease it.

2.2.3 Toroidal Embeddings Toroidal varieties (or schemes) étale-locally (or formally locally) are modelled on toric schemes. This turns out to be sufficient to extend various toric constructions, such as toric blowings up and toric resolution of singularities, to much wider context. The miracle enabling this is that these constructions pull back to the same operation on the toroidal scheme independently of the choice of a toric chart. For simplicity we will only consider the case of varieties. There is no torus action anymore, but it turns out that a large portion of the structure can be encoded just in the open orbit: Definition 2.9 (i) A toroidal variety (or a toroidal embedding) over a field k is a pair .(X, U ) with X a k-variety and .U → X an open subscheme such that étale locally X possesses an étale morphism to a toric scheme such that U is the preimage of the torus. Namely,  there exists an étale covering . i Xi → X and étale morphisms .hi : Xi → Tσi , called toroidal chart, such that .U ×X Xi is the preimage of the torus of .Tσi . If, moreover,

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the covering can be chosen to be Zariski, say .X = ∪i Xi , then the toroidal variety is called simple (or without self-intersections). (ii) A morphism of toroidal varieties .(Y, V ) → (X, U ) is any morphism .Y → X taking V to U . Example 2.10 If X is regular, D is a normal crossings divisor and .U = X \ D, then (X, U ) is a toroidal variety modelled on regular simplices .σ , and the free monoid .Mσ is generated by the regular parameters which define the branches of D (on an appropriate étale neighborhood).

.

2.2.4 (Non-)uniqueness of Charts A very natural question is to what extent the charts are unique. We will show below that the toric monoid is essentially unique, and the chart is unique up to units. For simplicity, × ∩ OX and .M = assume that .i : U → X is a simple toroidal variety and set .M = i∗ OU × M/OX (in the general case one would have to use étale sheaves, as we will do in the section about log schemes). The latter sheaf is the sheaf of toroidal Cartier divisors, i.e. divisors supported on .D = X \ U . Exercise 2.11 (i) Show that any local chart that maps x to the closed stratum of some .Tσ induces in isomorphism .Mσ = Mx , in particular, .Tσ depends only on .(X, U ) and x. (ii) Conversely, any section .s : Mx → Mx ⊂ OX,x of .Mx  Mx can be extended to a local chart on a neighborhood of x. (Hint: in addition to s one should choose a regular family of parameters on the stratum of .D = X \ U through x and lift them to elements of .OX,x .)

3

Logarithmic Structures and Schemes

In this section we introduce the category of log schemes and study its basic properties.

3.1

Monoids

Unless said to the contrary, by a monoid we always mean a commutative additively written monoid .M = (M, +, 0). By .M × we denote the subgroup of invertible elements and the sharpening of M is .M = M/M × . One says that M is sharp if .M = M.

3.1.1 Basic Constructions All categories of algebraic objects, such as groups, rings, commutative rings, etc. are complete and cocomplete—possess all small limits and colimits. Furthermore, limits

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are compatible with set-theoretic limits, and colimits are obtained using generators and relations. In particular, this is true for the category .Mon of monoids. The main examples of limits and colimits we will use are as follows: Exercise 3.1 (i) .M × N is just the usual product of sets with componentwise addition and it coincides with the coproduct, usually denoted .M ⊕ N . (ii) For homomorphisms .f : M → L, .g : N → L the fiber product .M ×L N is the submonoid of .M × N given by .f (m) = g(n). (iii) Instead of kernels in the category of monoids one uses congruence relations, that is, equivalence relations .R ⊆ M × M which are also submonoids: if .M → N is a surjective homomorphism, then the induced equivalence relation .R ⊆ M × M is a submonoid, and conversely any congruence relation appears in this way. (iv) For homomorphisms .f : L → M, .g : L → N the pushout .M ⊕L N is the quotient of .M ⊕ N by the minimal congruence relation such that .f (l) ∼ g(l) for any .l ∈ L.

3.1.2 Ideals An ideal .I ⊆ M is a subset such that .I + M = I , where the convention is that .I = ∅ is also an ideal (the analogue of the zero ideal of rings). An ideal I is prime if .x + y ∈ I implies that .x ∈ I or .y ∈ I . The set of all prime ideals is denoted .Spec(M) and called the fan of M. An ideal of the form .(a) = a + M is called principal. Exercise 3.2 Show that taking the preimage establishes a bijection between the ideals of M and .M.

3.1.3 Fine Monoids We will usually work with fine monoids: Definition 3.3 (i) M is finitely generated if there exists a surjective homomorphism .Nl → M. (ii) M is integral (or cancellative) if .n + m = n + m implies that .m = m . (iii) M is fine if it is integral and finitely generated.

3.1.4 The Grothendieck Group Recall that there is a canonical way to turn monoid into a group: Definition/Exercise 3.4 (i) Show that there is a universal homomorphism .M → M gp with .M gp a group, called the Grothendieck group of M. (Hint: for example, one can bound the cardinality of

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M gp because it is generated by the image of M, and then general representability theorems do the job because the category of groups is complete.) (ii) Construct .M gp explicitly as the quotient of .M 2 by the following equivalence relation: .(m, n) →(m , n ) if there exists .l ∈ M such that .l + m + n = l + m + n. (iii) Show that M is integral if and only if the homomorphism .M → M gp is injective. (iv) Define the integralization .M int to be the image of M in .M gp . Show that .M → M int is the universal homomorphism from M to an integral monoid and .M → M int is the left adjoint functor to the embedding of the category of integral monoids .Monint into .Mon. .

3.1.5 Fs Monoids An especially nice class of monoids is defined as follows: Definition 3.5 (i) An integral monoid M is saturated if for each .m ∈ M gp with .am ∈ M for .a ∈ Z>0 one has that .m ∈ M. (ii) A fine saturated monoid is called fs. (iii) A sharp fs monoid is called toric. Exercise 3.6 (i) Show that to give a toric monoid M is equivalent to give a lattice .M gp and a rational polyhedral cone .σ in .M gp ⊗ R such that .M = M gp ∩ σ . (ii) Show that .Spec(M) is naturally bijective to the set of faces of .σ . Exercise 3.7 Let .Monsat denote the category of saturated monoids. Show that the embedding .Monsat → Mon possesses a left adjoint functor, which is called the saturation functor and denoted .M → M sat . Show that .M sat is just the saturation of .M int in .M gp , that is the divisible hull of .M int in .M gp .

3.2

Logarithmic Structures

Definition 3.8 Let τ be one of the following topologies—Zariski, étale or flat. (i) A τ -prelogarithmic structure on a scheme X is a sheaf of monoids M on the site Xτ with a structure homomorphism of monoids u : M → (OX , ·). A homomorphism of prelog structures M → M is a homomorphism of sheaves of monoids compatible with the structure homomorphisms.

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(ii) A τ -logarithmic structure is a τ -prelogarithmic structure which induces an isomor× × × × ) →OX , and hence also M× → OX . The sharpening M = M/OX phism u−1 (OX τ τ τ τ is called the characteristic monoid of M. (iii) The default topology in this definition is the étale topology, so usually it will not be mentioned. A log structure M induces a Zariski log structure MZar just by restricting. By a slight abuse of language we say that M itself is Zariski if this restriction does not loose information, that is, M = ε∗ (MZar ) for the morphism of sites ε : Xet → XZar . Remark 3.9 (i) The homomorphism u is an analog of exponentiation, and this is one of the reasons to use additive monoids as the source. Traditionally it is denoted α, say α(m) = x, but probably the exponential notation, such as x = um instead of u(m), is more suggestive. Informally, any such m can be viewed as a branch of the logarithm of x, so the log structure can be viewed as fixing a monoid of branches of logarithms. (ii) The étale topology is used instead of the Zariski topology first of all in order to adequately treat toroidal embeddings, which are not strict. As a rule, this might pose mild technical inconveniences, which can be bypassed. For example, Kato restricted the generality to Zariski log structures in [6], but these results were generalized to the general case in [11]. (iii) Fppf log structures are sometimes needed to bypass positive characteristic problems, usually by extracting appropriate p-th roots. They are rarely used and will not show up in these notes. To get an initial feeling let us consider some examples. Example 3.10 × (0) The minimal or trivial log structure is just M = OX . et (1) The largest log structure with an injective u is M = OXet , but it is not too useful. The main exception is when X is a “small” scheme—a semi-local curve or the spectrum of a valuation ring, with the most useful case being when X is a trait. (2) A very important example of a log structure with an injective u : M → OX is as × follows. Assume that D → X is a closed subset and set M(logD) = OXet ∩ i∗ OU et where i : U → X is the open immersion of the complement U = X \ D. This is the log structure of elements invertible outside of D. Usually it is used when D is the underlying closed set of a Cartier divisor; in this case D is determined by the log structure and we call M(logD) a divisorial log structure. In particular, the toroidal scheme structure (X, U ) can also be encoded in the log structure of D-monomial elements, where D = X \ U . (3) The other extreme case is provided by so-called hollow log schemes with um = 0 for any m ∈ M \ M× . Usually they show up when one restricts a log structure on X

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onto a closed subscheme Z such that MX is generically non-trivial on Z. Often this is the most economical way to encode certain information about the ambient scheme X on Z. In particular, this is useful in deformation theory. In fact, it was such kind of an example, and not toroidal schemes, which led Fontaine and Illusie to introduce log schemes.

3.2.1 Associated Log Structure Any prelog structure .M can be canonically transformed into a log structure .Ma . The idea × × ) →OX so we should force this map is that .M is a log structure if and only if .u−1 (OX et et to be an isomorphism. This works for any topology, so for shortness we only consider the étale one. Exercise 3.11 Given a prelog structure .M on X let .Ma be the pushout of the diagram × M ← u−1 (OXet )× → OX . et

.

Show that .Ma is a log structure, .M → Ma is the universal homomorphism of .M to a log structure and the functor .M → Ma is left adjoint to the embedding of the category of log structures into the category of prelog structures. Remark 3.12 One can view the functor .M → Ma as an analogue of sheafification. Various operations on sheaves are defined by applying sheafification to the analogous operation in the category of presheaves (the sheafification is not needed for right exact functors, but is usually needed for other functors). In the same vein, various naive operations on log structures result in a prelog structure only (one works with sheaves, so the usual sheafification is used), and the functor .M → Ma should then be applied. Definition/Exercise 3.13 Integralization .Mint and saturation .Msat of a log structure .M are defined by applying the corresponding functors to monoids of sections, sheafification and then applying the functor a. Check that the resulting logarithmic structure is indeed saturated or integral.

3.2.2 Coherent Log Structures One reason to consider prelog structures was already discussed—they form a simpler category and various operations on log structures often have prelog structures as intermediate results. Another reason is that prelog structures are used as charts (see Sect. 3.3.3 below), usually of finite type, for log structures, which are often very large because of the invertible part.

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Definition 3.14 (i) A prelog structure is called constant if it is the sheafification of a homomorphism .P → (OX ) for a monoid P . For shortness we will not distinguish the monoid P and its sheafification. (ii) A log structure .M is called quasi-coherent if étale-locally there exists a constant prelog structure P such that .M = P a . If, in addition, P can be chosen finitely generated, .M is called coherent. We warn the reader that this notion is not related to coherence and quasi-coherence of .OX -modules. (iii) A coherent and integral (resp. saturated) logarithmic structure is called fine (resp. fs). The following result can be found, for example, in [13, Corollary II.2.3.6] Exercise 3.15 Show that a log structure is fine (resp. saturated) if and only if étale locally it possesses a fine (resp. fs) chart .P → OX . Finally, for a morphism .f : Y → X one defines direct and inverse images of the log structures .M on X and .N on Y : Definition/Exercise 3.16 (i) Show that .f∗ (N ) ×f∗ (OY ) OX is a log structure denoted (by a slight abuse of notation) .f∗ (N ). (ii) The inverse image .f ∗ (M) is the log structure associated to the prelog structure −1 (M) → f −1 (O ) → O . Show that, as expected, .f ∗ is left adjoint to .f . .f X Y ∗

3.3

Logarithmic Schemes

Now we can introduce the category of log schemes. Definition 3.17 (i) A log scheme X is a tuple .(X, MX , uX ), where .X is a scheme called the underlying scheme and .uX : MX → OX is a log structure on X. Usually we will use looser notation when this cannot lead to a confusion, e.g. write u instead of .uX or just omit it, write .OX instead of .OX or even denote the underlying scheme by the same letter X. (ii) A log scheme X is called quasi-coherent, coherent, integral, fine, fs, etc., if the log structure .MX is quasi-coherent, coherent, integral, fine, fs, etc.

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(iii) A morphism of log schemes .f : Y → X consists of the underlying morphism ∗ .f : Y → X of schemes and a homomorphism of log structures .f MX → MY compatible with the structure homomorphisms .uX and .uY . Remark 3.18 (i) We warn the reader that the word “integral” becomes slightly overused, because the notion of integral schemes means something completely different in the theory of schemes. So one should use it carefully, to avoid misunderstandings. Typically, if needed, one stresses that the underlying scheme is integral. (ii) Already in [5] Kato noticed that non-integral log schemes are too pathological and mainly restricted consideration to the category of fine log schemes. Although some aspects of a more general theory were developed quite systematically in [13], most of studies are done in the generality of fine log schemes, and this indeed seems to be the best choice. The second popular choice, which is often used once one works with log blowings up, is to work with the subcategory of fs log schemes. One benefit of working with these categories is that integral or saturated fiber products often reveal a nicer behaviour. In particular, log blowings up are log étale monomorphisms, see Exercise 4.20 and this fact essentially uses integralization. Remark 3.19 (i) In case of a log structure with an injective .MY → OY a morphism .f : Y → X is uniquely determined by the underlying morphism of schemes, and .f extends to f if the log structure on Y is “larger” than the image of the log structure on X. If .MY = M(logE) and .MX = M(logD) are divisorial, then this happens if and only if .f −1 (D) ⊆ E. (ii) In particular, providing a toroidal scheme .(X, U ) with the divisorial log structure .MX = M(log(X \ U )) we obtain a fully faithful embedding of the category of toroidal schemes into the category of log schemes. Naturally, such .(X, MX ) will be called a toroidal log scheme. (iii) The other extreme is when the log schemes are hollow: .uX = 0, .uY = 0. In this case, ∗ .f extends to f via any homomorphism .f (MX ) → MY .

3.3.1 Strict Morphisms A very important class of morphisms are those that “minimally modify the log structure”. In the case of divisorial log structures this just means that .E = f −1 (D) and in general: Definition 3.20 A morphism .f : Y → X of log schemes is called strict if .f ∗ (MX ) = MY .

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In particular, for a log scheme X any morphism .Y → X can be uniquely enhanced to a strict morphism of log schemes. g

h

Remark 3.21 Any morphism .Y → X canonically factors into a composition .Y → Z → X such that h is strict and .Y = Z. It is natural to view h as a scheme-like morphism and g as a morphism which only increases the log structure. However, this factorization is not especially useful. The reason is that g is not a “monoidal-like” morphism, see Sect. 4.1.1 below. Example 3.22 (i) In the category of log schemes strict closed immersions play the role of usual closed immersions in the category of schemes. Often strict closed immersions have hollow sources. A typical example is when X is a toroidal log scheme and .Y → X is a toroidal subscheme. (ii) The simplest and most important case is when X is the affine line with marked origin (i.e. .X = Spec(k[t]) and .MX = (N log t)a , where we denote the generator of the monoid by .log t to stress that it is mapped to t by .uX ) and .y = Spec(k[t]/(t)) is the origin. The induced log structure is .My = k × ⊕ N log t and we call .(y, My ) the standard log point. It is an analogue of points with non-reduced scheme structure in the usual algebraic geometry, in particular, we will later see that it is not log smooth and has non-trivial log differentials coming from the log direction .log t. Note also that any other point .x = Spec(k[t]/(t − a)), .a ∈ k × has the trivial induced log structure since .N log t is mapped to .Ox× and hence the functor .M → Ma shrinks .k × ⊕ N log t to .Mx = k × by sending .log t to a. (iii) Similarly, for each .n ≥ 1 the thick point .Y n = Spec(k[t]/t n+1 ) can be provided with the log structure induced by .N log t. It is neither injective, nor hollow. (iv) One can also consider other log points, for example, for a toric monoid P the log 0

structure induced by .P → k corresponds to the origin of the toric scheme .Spec(k[P ]).

3.3.2 Log Rings For a local work with log schemes it is often convenient to use the following logarithmic spectrum construction. Definition 3.23 (i) A log ring .(A, P , u) consists of a ring A, a monoid P and a homomorphism .u : P → (A, ·). u (ii) The logarithmic spectrum of a log ring, usually denoted just by .X = Spec(P → A) is the underlying scheme .X = Spec(A) provided with the log structure induced by the prelog structure u.

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Remark 3.24 (i) In a sense this is a prelog ring, but the notion of a log ring does not make too much sense—even if .P = (MX ), this condition will be lost after localizations. (ii) A log scheme is Zariski if and only if it is a spectrum of log rings Zariski-locally. For general log schemes this is only true étale-locally, so in this aspect they are analogous to algebraic spaces.

3.3.3 Charts The notion of charts of log structures can be adopted to the category of log schemes once one replaces a monoid P by the associated log scheme. Definition/Exercise 3.25 (i) For a monoid P let .AP denote the log scheme whose underlying scheme is .Spec(Z[P ]) and the log structure is induced by the homomorphism .P → Z[P ]. If we work over a base scheme, for example, .S = Spec(k), then we will use the notation .AS,P = S[P ] := S × Z[P ]. (ii) Let X be a log scheme. Show that giving a global chart .φ : P → OX for the log structure .MX is equivalent to giving a strict morphism of log schemes .f : X → AP . Any such morphism f is called a global chart of X, and we will not make a real distinction between two presentations of a chart. A chart is called fine, saturated, sharp, etc. if the monoid P is fine, saturated, sharp, etc. In particular, check that a coherent (resp. fine, resp. fs) log scheme is a log scheme which étale-locally possesses a finitely generated (resp. fine, resp. fs) chart. Remark 3.26 It is important to consider charts with non-sharp P because various operations can produce non-sharp monoids. For example, removing the origin from .S[N] = Spec(k[t]), where .S = Spec(k), one obtains the scheme .S[Z] = Spec(k[t ±1 ]), which is also a chart of itself. On the other hand, its log structure is trivial, hence .S[Z] → S is also a chart. Nevertheless, when working locally at a point one might want to take a smallest possible chart and often this is possible. The following notion is due to Kato: Definition 3.27 A chart .P → OX is called neat at a geometric point .x → X if .P →  Mx . In particular, P is automatically sharp. Example 3.28 Let .X = Ak,N = Spec(k[t]). Then the tautological chart .N log t → OX is neat at the origin, but not at the other points, where a smaller chart (the trivial one) exists.

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The following example demonstrates one of rare aspects, in which fppf fine log schemes behave nicer than their étale analogues. However, even this is only needed when the log schemes are only fine but not fs. The following results can be found, for example, in [13, §II.2.3]. Exercise 3.29 Let X be a fine log scheme, let .x → X be a geometric point and let P = MX,x . Any neat chart at .x gives rise to a section .P → MX,x of the sharpening homomorphism.

.

(i) Show, that conversely any section .s : P → MX,x of .MX,x → P induces a chart for a small enough étale neighborhood of .x and this chart is neat at .x. (ii) Furthermore, show that such a section exists if and only if the sequence × → MX,x → P gp → 0 1 → OX et ,x

.

gp

splits, and this is automatic whenever .P gp has no torsion of order divisible by .p = char(k(x)) [13, Proposition II.2.3.7]. (iii) Show that any fine fppf log scheme admits neat charts fppf-locally. (Hint: using the fppf topology one can also extract roots of order p.) (iv) Show that if the log structure is fs, then .P gp is automatically torsion free, and hence even Zariski log schemes possess neat charts locally at points of .X.

3.3.4 Monoidal Morphisms We say that a morphism of log schemes .Y → X is monoidal if étale-locally on X it is the base change of morphisms of the form .AQ → AP . Informally speaking, Y is obtained by first changing the monoidal structure and then adjusting the underlying scheme in the minimal needed way. Also, such morphisms can be viewed as base changes of morphisms of fans of monoids, e.g. see the informal notation [6, Definition 9.10]. Main examples of such morphisms are integralization, saturation, log blowings up and the morphism .LogX → X. They all will be discussed later, and we start with the first two. Exercise 3.30 (i) Show that for any coherent log scheme X there exists a universal morphism .X → Xint (resp. .X → Xsat ) whose target is a fine (resp. fs) log scheme. In other words, the functor .X → Xint (resp. .X → Xsat ) is left adjoint to the embedding of the category of fine (resp. fs) integral schemes into the category of coherent log schemes. In addition, int is a closed immersion and .X → X sat is finite. (Hint: first, assuming that .X → X X possesses a chart .X → Spec(Z[P ]) show that .Xint = X ⊗Z[P ] Z[P int ] and .Xsat = X ⊗Z[P ] Z[P sat ] are as required. In general, use local-étale charts and étale descent of finiteness.)

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(ii) Show that for any strict morphism of coherent schemes .Y → X one has that .Y int = Xint ×X Y and .Y sat = Xsat ×X Y .

3.3.5 Fiber Products Similarly to the category of schemes, the category of log schemes and its subcategories possess all finite limits, of which we will use only fiber products. Exercise 3.31 (i) Let .{Xi }i∈I be a finite diagram of log schemes. Show that .X = limI Xi exists and can be described as follows: .X = limI X i and the log structure .MX is the colimit of the pullbacks of .MXi to .MX . (In particular, this involves the functor a; the shortest way is to pullback .MXi as prelog structures, then take the colimit, and then apply a once.) (ii) Show that if all .Xi are fine or fs, then there exists a limit in the same category, and it coincides with .Xint or .Xsat , respectively. Remark 3.32 Often one mentions .int or .sat in the notation of the limit, e.g. .Y ×sat X Z or int .(Y ×X Z) , to avoid confusions. Sometimes, when only fs (resp. fine) log schemes are considered, this superscript can be omitted. To feel how this works, we start with the following almost tautological fact. Exercise 3.33 If S is a scheme, .P → Q, .P → R are homomorphisms of finitely generated monoids and .X = S[P ], .Y = S[Q], .Z = S[R], then .Y ×X Z = S[Q ⊕P R], int = S[(Q ⊕ R)int ] and .(Y × Z)sat = S[(Q ⊕ R)sat ]. .(Y ×X Z) P X P Now we can consider an archetypical example of a subtle behaviour of fine log schemes, as opposed to coherent log schemes or schemes. Exercise 3.34 (i) Let .X = Spec(k[x, y]) with the log structure induced by .Nlog(x) ⊕ Nlog(y), and let y .Y = Spec(k[x, ]) with the log structure induced by .Nlog(x) ⊕ N(log(y) − log(x)). x 2 Show that .Y = Y ×int X Y , in particular, the integralization functor cuts off the .A component from .Y ×X Y . (ii) More generally, assume that .P  Q are toric monoids such that .P gp = Qgp . Then int A , that is, the morphism .A .AQ = AQ × Q Q → AP is a monomorphism in the AP category of fine log schemes, but not in the category of coherent log schemes. Remark 3.35 Usual blowing up have some nasty properties. For concreteness, consider the blowup up chart .f : Y = Spec(k[x, y  = yx ]) → X = Spec(k[x, y]) at the origin. Clearly, f is non-flat, the dimension of the fiber over the origin jumps, and this

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even gives rise to the new irreducible component .Spec(k[y1 , y2 ]) in .Y ×X Y . Assume now that .X and .Y are provided with the log structures generated by .log(x), log(y) and .log(x), log(y  ). Then the .A2 component in the product acquires a non-cancellative log structure with monoid presented by generators .log(x), log(y1 ), log(y2 ) and relation   .log(x) + log(y ) = log(y) = log(x) + log(y ). This forces the integralization functor to 1 2 2 remove the .A component and only its diagonal, which is the intersection with the other component, is left. In fact, we will later see that many similar morphisms (log blowings up and their charts) are monomorphisms in the fine category. And here is an archetypical example of a property of the saturated category. It explains why one usually restricts the setting to fs log schemes when studying Kummer covers. Exercise 3.36 (i) Let .X = Spec(k[x]) with the log structure .N log(x) and .Y = Spec(k[x 1/2 ]) with the log structure . 12 N log(x). We will later see that the Kummer cover .Y → X is log étale when .char(k) = 2. Check that .Z = Y ×X Y = (Y ×X Y )int contains two components (diagonal and antidiagonal) intersecting over the origin and the characteristic .Mz at the intersection point z is the non saturated monoid . 12 N ⊕N 12 N with generators 1 1 .( , 0), (0, ) subject to the relation .(1, 0) = (0, 1). It is isomorphic to the submonoid 2 2 of .Z/2Z ⊕ N obtained by removing the element .(1, 0). (ii) Show that .Z sat is the normalization of Z and it is just the disjoint union of the two copies of Y —the diagonal and the antidiagonal. The characteristic of the two points sat Z/2Z ⊕ N. over z is .N—the sharpening of .Mz → (iii) More generally, if S is a scheme, .P ⊆ Q is a Kummer extension of toric monoids, i.e. Q is the saturation of P in .Qgp , and .X = S[P ], Y = S[Q], then .Z = Y ×sat X Y gp gp is isomorphic to the split cover .Y × G for .G = Q /P . So, the log étale G-Galois cover .S[Q] → S[P ] behaves similarly to étale covers only in the category of fs log schemes.

3.4

Logarithmic Regularity

In this section we only consider fs log schemes.

3.4.1 The Logarithmic Stratification gp Each log scheme X possesses a natural stratification by .rk(Mx ) = dimQ (Mx ⊗ Q).

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Exercise 3.37 (i) Let .C≥d ⊆ X be the locus on which the rank of the characteristic monoid is at most d. Show that these sets are closed and hence induce a stratification by the locally closed sets .Cd = C≥d \ C>d . (ii) Refine this stratification to a log stratification .Xd of X by (non-necessarily reduced) locally closed subschemes as follows: if the structure is Zariski at x and .d = rk(Mx ), + then the stratum .X ≥d at x is given by the vanishing of .uMx , where .M+ x is the maximal ideal of .Mx . Show that this is compatible with strict morphisms and hence extends to arbitrary log structures by étale descent. Finally, set .Xd = X≥d \ X>d . (iii) Show that, indeed, .C≥d is the reduction of .X≥d . Also, show that the log strata of the chart log schemes .AP are reduced.

3.4.2 Logarithmic Regularity The following definition is a far-reaching generalization of the classical fact recalled in Exercise 2.2(iii). Definition 3.38 A locally noetherian fs logarithmic scheme X is logarithmically regular if each locally closed subscheme .X r = X ≥r \ X≥r+1 is regular (in particular, reduced) and of codimension r. Remark 3.39 The original definition by Kato only considered fs log schemes. Most of results about log regularity can be extended to fine log schemes, and this was worked out by Gabber, but we will not touch this direction in the notes.

3.4.3 Log Parameters As in the case of regular schemes, when working with log regular log schemes it is very convenient to use local parameters. The classical notion is generalized as follows: Definition 3.40 Let X be a log regular log scheme with a geometric point .x → X over x ∈ X. Let .r = rk(Mx ). By a regular family of parameters at .x we mean a section .s : Mx → Mx and elements .t1 , . . . , td ∈ Ox , such that .t1 , . . . , td restrict to a regular family of parameters of .Xr at x. In particular, d is the dimension of .Xr at x and .r + d = dimx (X). We call .ti regular parameters and the elements .us(m) for .0 = m ∈ Mx will be called logarithmic parameters.

.

In view of Exercise 3.29(iv) such families of parameters exist, and if the log structure at x is Zariski, one can even construct s Zariski locally. As in the classical case, it follows from the definition that any regular family of parameters generates the maximal ideal at .x. Furthermore, parameters naturally give rise to very explicit étale and formal charts.

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Exercise 3.41 Assume that a log scheme X is Zariski and log regular at a point .x ∈ X with P = Mx , and let .s : P → Mx and .t1 , . . . , td ∈ Ox be a regular family of parameters at x. If .k = k(x) is of positive characteristic p, let .Ck be a Cohen ring of k, that is a DVR with maximal ideal .(p) and residue field k. By Cohen’s theorem if X is of equal characteristic x , while in the mixed characteristic at p, then there exists a field of coefficients .i : k → O x . Prove the following theorem of Kato case there exists a ring of coefficients .i : Ck → O (see [6, Theorem 3.2]), where .AP  denotes the formal completion of .A[P ] at the ideal + .A[P ]:

.

x , (i) In the equal characteristic case the natural homomorphism .kP t1 , . . . , td  → O induced by i and the parameters, is an isomorphism. x induced by (ii) If .char(k) > 0, then the natural homomorphism .Ck P t1 , . . . , td  → O i and the parameters is surjective with a principal kernel .(θ ), where .θ ≡ p modulo + 2 .(P , t1 , . . . , td , p) . Remark 3.42 (i) The equal characteristic p case naturally shows up in both cases, since one can take .θ = p. (ii) In the equal characteristic case Kato’s theorem tells that log regularity is the same as being formally-locally isomorphic to a toric variety. So, the theory of log regular schemes can be viewed as a generalization of toroidal geometry to the mixed characteristic case. (iii) A relatively difficult theorem asserts that log regularity is preserved by localizations. Kato’s original proof is incomplete, but Gabber later provided missing arguments. The source of the difficulty is clear—regular parameters at generizations of x are not related to parameters at x. In fact, this is completely parallel to the situation with usual regularity. However, in the classical case there is a conceptual proof which uses Serre’s cohomological criterion of regularity, and no logarithmic analogue was found so far.

3.4.4 Log Regularity and Toroidal Varieties Finally, let us describe log regular log varieties over a field k. As in the case of usual varieties, a nice description (in terms of smoothness) is possible only for simple points x, that is, points for which the extension .k(x)/k is separable. Exercise 3.43 Assume that X is a log variety over .S = Spec(k) and .x ∈ X is such that k(x)/k is separable and X is log regular at x. Let .x → X be a geometric point over x.

.

(i) Prove that locally at .x there exist a chart .f : U → S[P ⊕ Zd ] with an étale f , where .P = Mx and d is the dimension of the logarithmic stratum at x. Deduce that X is

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toroidal at .x, and thus being log regular and toroidal at a simple point are equivalent. (Hint: use parameters and work with étale topology instead of the formal one.) (ii) Prove that any chart .f : U → S[P ], which is neat at .x, is smooth at the image of .x.

4

Morphisms of Logarithmic Schemes

Our next goal is to study morphisms of log schemes in more detail.

4.1

Charts

Recall that charts of log schemes were defined in Definition 3.25. Naturally, by a chart of a morphism of log schemes one means compatible charts of the source and the target: Definition 4.1 Let .f : Y → X be a morphism of log schemes. A chart of f consists of charts .Y → AQ and .X → AP and a homomorphism .φ : P → Q such that the compositions .P → (MX ) → (MY ) and .P → Q → (MY ) coincide. Equivalently, the chart is a commutative diagram

.

whose horizontal lines are charts. One says that the chart is modeled on .φ. It is easy to construct charts, morally, one just starts with a chart .P → X and enlarges it by adding enough elements of .MY . Exercise 4.2 Let .f : Y → X be a morphism of fine log schemes, .x → X a geometric point and .y → Y a geometric point above .x. Prove that any étale-local fine chart .P → Mx at .x extends to a chart of f étale-locally at .y. (Hint: start with any fine chart .Q → My and take Q to be the image of .P ⊕ Q → My .)

4.1.1 The Standard Splitting If a log scheme X is provided with a chart .X → AP and .P → Q is a homomorphism of monoids we will use the notation .XP [Q] = X ×AP AQ , which indicates that .XP [Q] is obtained by a “base change of the monoidal structure”. A chart of f induces a very useful splitting of f into the composition .Y → XP [Q] → X, where the first morphism is strict, as .AQ is a chart of both the source and the target, and the second morphism is monoidal. In a sense, this decomposes f into a composition of a scheme-like morphism and a monoidal-

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like morphism, and such a splitting is ubiquitous in log geometry. However, it is noncanonical and exists only étale-locally.

4.1.2 Neat Charts As with the neat absolute charts, to efficiently work with charts of morphisms one would like to construct minimal charts, or even a minimal chart extending a given neat chart of the target. It turns out that in general one cannot construct charts modeled on homomorphisms of sharp monoids, as we are going to demonstrate. Let .f : Y → X be a morphism of Zariski log schemes, .y ∈ Y , .x = f (y), .Q = My and .P = Mx . The induced homomorphism .φ : P → Q has no kernel, since any non-unit of .Mx is mapped to a non-unit in .Ox and hence also to a non-unit in .Oy . Nevertheless, it might happen that .φ is not injective, but .Ker(φ gp ) ∩ P = 0. Here is an archetypical example: Example 4.3 Let .X = Spec(k[s, t]) with the log structure generated by s and t, and .Y = Spec(k[s, u]) with .u = t/s a chart of the blowing up of X at the origin .x ∈ X with the log structure generated by s and u. Let .y ∈ Y be a point given by .s = 0, .u = a ∈ k × . Then × .P = Mx = N log(s) ⊕ N log(t) and .Q = My = N log(s) because .u ∈ Oy . The map .P → Q sends .log(t) to .log(s) and .log(u) = log(t) − log(s) generates the kernel of the map .P gp → Qgp . The above example motivates the following definition with the idea to provide Q with the minimal amount of units so that the homomorphism .P → Q is injective. Definition 4.4 (i) The relative characteristic monoid of .f : Y → X is .MY /X = Coker(f ∗ (MX ) → MY ). Note that it also coincides with .Coker(f −1 (MX ) → MY ), hence .MY /X = MY /X . (ii) Assume that .Y → AQ , .X → AP , .Aφ is a chart of a morphism of log schemes .f : Y → X, and .y ∈ Y is a point with .x = f (y). The chart is called neat at y if .φ is gp injective and the induced homomorphism .Coker(φ gp ) → MY /X,y is an isomorphism. Neat charts always exist in fppf topology, while in étale topology there might be an obstacle if .Coker(φ gp ) has a p-torsion for .p = char(k(y)). Proof of the following result, which can be found in [13, Theorem II.2.4.4], involves a bit of diagram chasing with monoids and their Grothendieck groups. Exercise 4.5 Assume that .f : Y → X is a morphism of log schemes, .y → Y a geometric point whose image in X is .x ∈ X. Assume that .h : U → AP is a neat chart of a gp neighborhood of x. If .Ext1 (MY /X,y , Oy× ) = 0, then locally along .y there exists a neat chart of f that extends h. In particular, a neat chart exists when the order of torsion of gp .Coker(M Y /X,y ) is invertible in .k(y).

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Remark 4.6 The same argument proves that neat charts always exist fppf locally. Simigp larly, if the log structure at y is Zariski and .MY /X,y is torsion free, a neat chart exists Zariski locally, but if the relative characteristic monoid contains a torsion, one might need to extract roots of some units, ending up with an étale-local or fppf-local chart—depending on the torsion.

4.2

Logarithmic Smoothness

4.2.1 Log Thickenings The following definition is a direct extension of its scheme-theoretic analogue. Definition 4.7 (i) A log thickening is a strict closed immersion .S → T given by a nil ideal .I ⊂ OT . (ii) A morphism of log schemes .f : Y → X is called formally log smooth (resp. formally log étale, resp. formally log unramified) if for any log thickening and compatible morphisms .i : S → Y , .T → X étale locally on T there exists (resp. there exists unique, resp. there exists at most one) lifting .T → X making the diagram commutative:

.

(iii) A morphism f is log unramified if it is formally log unramified and .f is of finite type. A morphism f is log smooth (resp. log étale) if it is formally log smooth (resp. log étale) and .f is finitely presented. Similarly to the theory of schemes one can study these notions most effectively by use of log differentials.

4.2.2 Logarithmic Derivations Log geometry has a version of the theory of derivations and differentials. The idea is to extend the usual theory by adding logarithmic differentials of monomials. Definition 4.8 Let .A → B be a homomorphism of log rings .A = (P → A) and .B = (Q → B) and let N be a .B-module. An A-log derivation .(d, δ) : B → N consists of an .A-derivation .d : B → N and a homomorphism .δ : Q → N such that .δ(P ) = 0 and q q .u δ(q) = d(u ). The .B-module of all log derivations will be denoted .DerB/A (E). Remark 4.9 One should view .δ as derivation of the branch of the logarithm of the monomial .uq specified by q.

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4.2.3 Logarithmic Differentials Similarly to usual Kähler differentials, there always exists a universal log derivation whose target is the module . 1B/A of logarithmic differentials. This is not so standard, but we omit 1 in the notation because we will never consider modules of differential p-forms with .p > 1. Exercise 4.10 Prove that the functor .DerB/A (·) is representable and describe the corresponding universal module as follows: . 1B/A is the quotient of . 1B/A ⊕ (B ⊗ (Qgp /P gp )) by the submodule generated by the relations .(duq , −uq ⊗q) with .q ∈ Q, where we denote the image in .Qgp /P gp also by q. We will not go into details, but most of classical results about Kähler differentials, such as base change, compatibility with localizations and fundamental sequences, extend to log rings and log differentials. In addition, the module of differentials of any strict étale homomorphism vanishes, hence by use of étale descent the definition can be globalized to the definition of an .OY -module . 1Y /X for any morphism .Y → X of log schemes, and it comes equipped with the universal log derivation .(d, δ) : (MX → OX ) → 1Y /X . In particular, d induces a homomorphism . 1Y /X → 1Y /X . As usual in log geometry, to get some feeling of a new notion one should look at the two polar cases, and the case of strict morphisms reduces to the usual scheme theory. Example/Exercise 4.11 Use the previous exercise to prove that: (i) If .Y → X is a strict morphism of log schemes, then . 1Y /X = 1Y /X . (ii) If R is a ring, .Y = AR,P , .Y = AR,Q and .f = AR,φ for .φ : P → Q, then 1Y /X = Coker(φ gp ) ⊗ R[Q].

.

In particular, if .Q ⊆ R, then . 1Y /X is free and its rank is the rank of .Coker(φ gp ), and if .Fp ⊆ R, then . 1Y /X is free and its rank is the p-rank of .Coker(φ gp ). (Hint: show that .DerR[Q]/R[P ] (E) = Hom(Coker(φ gp ), E).) 0

(iii) By a slight abuse of notation for a sharp monoid P and a ring R we denote by .P → R the homomorphism of monoids taking .P + to 0 (but 0 goes to 1). Compute the 0

differentials of a log point .Spec(P → k). More generally, show that for any ring 0

R and the hollow log scheme .X = Spec(P → R) there is an isomorphism of Rmodules . 1X/R = P gp ⊗ R.

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4.2.4 Chart Criterion The main theorem of Kato about logarithmic smoothness gives the following criterion for log smoothness in terms of charts, see [5, Theorem 3.5]: Theorem 4.12 For a morphism .f : Y → X of fine log schemes and a geometric point y → Y the following conditions are equivalent:

.

(i) f is log smooth (reps. log étale) at .y, (ii) Locally at .y there exists a chart .V → AQ , .U → AP modelled on .φ : P → Q such that (a) The morphism .V → UP [Q] is smooth (resp. étale) (b) .Ker(φ gp ) and .Coker(φ gp )tor (resp. .Ker(φ gp ) and .Coker(φ gp )) are finite of order invertible in .k(y). (iii) The same condition as (ii) but with .V → UP [Q] étale. Exercise 4.13 (i) Check that the condition on .φ is equivalent to smoothness (resp. étaleness) of the morphism of diagonalizable groups .Dk(y),Qgp → Dk(y),P gp and by the above criteria it is also equivalent to log smoothness (resp. log étaleness) of the map .Ak,Q → Ak,P . (ii) Deduce (iii) from (ii) by enlarging the chart to .Q ⊕ Zn , where .t1 , . . . , tn are regular parameters on the fiber of .V → UP [Q] through .y. (Hint: for example, one can send the generators of .Zn to .1+ti (i.e. one increases Q by adding the elements .log(1+ti ).) (iii) Check that in condition (b) one can also achieve that .Ker(φ) = 0 (Hint: again, just increase Q accordingly.) Here as a very typical example involving log points. We will return to this setting also in Example 4.23. Example 4.14 0

(i) Show that log points .Spec(P → k) with a sharp fine .P = 0 are not log smooth over k. (ii) Take .P = N log(t) and consider the nodal curve .C = Spec(k[x, y]) with the log structure induced by .Q = N log(x) ⊕ N log(y). Show that the morphism .C → P taking .log(t) to .a log(x) + b log(y) with .a, b ∈ N is log smooth if and only if .(a, b) is invertible in k. Show that if .(a, b) ∈ k × , then (unlike . 1C/k ) the module . 1C/P is invertible; in fact, it is generated by .δ(x), δ(y) subject to the relation .aδ(x)+bδ(y) = 0. (iii) Give another proof of the results of (ii) by noting that the morphism .C → P is just the fiber over the origin of the morphism .AQ → AP given by .t = x a y b .

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An important theorem of Kato states that log regularity is preserved by log smooth morphisms, see [6, Theorem 8.2]: Theorem 4.15 If X is a log regular log scheme and .Y → X is log smooth, then Y is log regular.

4.3

Logarithmic Blowings Up

4.3.1 Log Ideals Definition 4.16 Let X be a fine log scheme. By a log ideal we mean a coherent sheaf of ideals J ⊆ MX , where coherence means that locally around any geometric point x the ideal is generated by Jx . A log ideal is called invertible if it is locally generated by a single element. Remark 4.17 Since monoids MX (U ) are integral there is no need to impose a condition analogous to being a non-zero divisor in a ring, and invertible ideals are preserved by arbitrary pullbacks, unlike the theory of schemes.

4.3.2 Log Blowings Up Log blowings up are defined by the same universal property as usual blowings up, but with log ideals used instead of ideals. Definition 4.18 Let X be a log scheme and J a log ideal on X. A morphism .f : Y → X is called the log blowing up of X along J and denoted .LogBlJ (X) → X if f is the universal morphism of fine log schemes such that .f −1 (J ) is invertible. The saturated log blowing up is defined by the same property but in the category of fs log schemes. Thus, it is nothing else but the saturation of the log blowing up. Exercise 4.19 Log blowings up are preserved by any base changes .h : X → X, i.e.  = LogBl   .Y h−1 J (X ) = Y ×X X . In particular, applying this to Y one obtains that .Y ×X Y = Y and hence .Y → X is a monomorphism. Now we are going to prove that log blowings up exist and are, in fact, monoidal morphisms, as one might expect, since they realize a monoidal construction. Exercise 4.20 We will construct .Y = LogBlJ (X) in a few steps. (i) Prove that if .X = AP and .I = Z[J ], then .Y = BlI (X) and the log structure on the schart .Y s = Spec(Z[P ][ Is ]) of the blowing up is given by .P [J − s] (the submonoid of gp generated by P and the elements of .J − s). Furthermore, the chart .Y of Y is the .P s

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universal log scheme over X such that the pullback of J to .Ys is principal generated by s. (ii) Prove that if X possesses a chart .X → AP and J is generated by an ideal .J0 ⊆ P , then .LogBlJ (X) = LogBlJ0 (AP ) ×AP X. Deduce that log blowing up .Y → X always exists and is a proper monoidal morphism. (Hint: use part (i), Exercise 4.19 for the first claim and then apply étale descent.) (iii) Prove that any log blowing up is a log étale morphism. (Hint: by base change and étale descent this reduces to the case described in (i) and then Kato’s criterion does the job.) (iv) Finally, reinterpret examples we have seen: using Exercise 3.34(ii) prove directly that .Y = Y ×X Y , and explain which chart of a log blowing up is considered in Remark 3.35. Typically, one asks when a log blowing up coincides with the usual blowing up along the ideal .uJ OX generated by J , because this seems to be the most adequate situation. Clearly, this happens if and only if the relevant toroidal blowings up of the charts .AP are compatible with the chart maps .h : U → AP . In particular, this is automatic if h is flat or, at least, what is called .Tor-independent from J , that is, .Tor1Z[P ] (OX , Z[P ]/J n Z[P ]]) = 0. Using the latter criterion Nizioł proved the following claim, see [11, Proposition 4.3]: Theorem 4.21 If X is a log regular log scheme and J is a log ideal, then the underlying scheme of the .LogBlJ (X) is the blowing up of X along the induced ideal .uJ OX . In addition, the saturated log blowing up .LogBlJ (X)sat is log regular and the log structure is induced by the union of the preimage of the toroidal divisor on X and the exceptional divisor. In fact the approach with Tor functors is only needed in the more difficult case of mixed characteristics. Exercise 4.22 Prove Theorem 4.21 when X is equicharacteristic. (Hint: formal completions of noetherian schemes are flat and hence compatible with blowings up. Formally locally X looks as .Spec(kP t1 , . . . , td ), hence the blowing up is described easily via the base change from the toric case via the flat homomorphism .k[P ] → kP t1 , . . . , td . The log regularity follows from Theorem 4.15.) Now let us discuss a few cases, that are simpler to compute but are often viewed as pathological and not worth consideration. In particular, one can easily have that the proper morphism .LogBlJ (X) → X is not birational. Using such morphisms becomes critical if one wants to study relative desingularization over singular bases (e.g. the log point or thick log point with a non-reduced scheme structure), but this direction has not studied yet in the literature, and we will only mention it in a couple of remarks later.

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Example/Exercise 4.23 0

(i) Show that the log blowing up of the ideal .P + on the log point .Spec(P → k) modeled on a fine sharp monoid P is of dimension .rk(P ) − 1. 0

(ii) Let .s = Spec(N log(t) → k) be the standard log point (i.e. .t = 0 in k) and Cs = Spec(N log(x) ⊕ N log(y) → k[x, y]/(xy))

.

a log smooth s-curve with .log(t) mapped to .log(x) + log(y). Let J be the maximal ideal of .N log(x)⊕N log(y). Show by a direct computation that .Xs = LogBlJ (Cs ) is also log smooth over s, the map .Xs → Cs is an isomorphism over the complement of the origin .O ∈ Cs and the preimage E of O is a non-reduced double .P1k component with the nilpotent ideal .(ε) and the log structure given by .N log(ε) and .log(t) mapping to .2 log(z). (iii) Now, embed s as the origin of .S = Spec(N log(t) → k[t]) and .Cs as the closed fiber of a log smooth (even semistable) S-curve C = Spec(N log(x) ⊕ N log(y) → k[x, y])

.

with .t = xy. Recall that .LogBlJ (Cs ) is the pullback of the log scheme .LogBlJ (C), whose underlying scheme is just .BlO (C). Use this to conceptually explain the results of (ii), in particular, the reason why the new component is doubled (has a non-reduced structure).

4.4

Logarithmic étaleness

In this section we restrict to the fs setting, in which Kummer covers are usually studied. We just give definitions, check simplest properties and mention various directions studied in the literature.

4.4.1 Kummer étale Morphisms A homomorphism of toric monoids .φ : P → Q is called Kummer if .P gp ⊆ Qgp is of finite index and Q is the saturation of P in .Qgp . A log étale morphism of log schemes .Y → X is called Kummer if the induced homomorphisms of monoids .Mx → My are Kummer. A Kummer étale cover .Y → X is a surjective Kummer étale morphism. Exercise 4.24 Check that, indeed, this notion of a covering defines a Grothendieck topology called Kummer étale of .ket topology. (Hint: this mainly reduces to the check that Kummer étale covers are preserved by base changes.)

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Remark 4.25 (i) Kummer étale topology is the closest analogue of the étale topology in the setting of fs log schemes. For example, see Exercise 3.36. An important fact is that the theory of descent works pretty similarly to the case of étale (or flat) topology: .OXket and, more generally, representable functors are sheaves (see [12, Proposition 2.18 and Theorem 2.20]), etc. Ideals in .OXket are called Kummer ideals. Working with them provides a convenient formalism for extracting roots from monomials. (ii) Theories of Kummer étale and general log étale cohomologies are now developed rather deeply, see [9] and [10]. The first one is simpler, but in order to have some fundamental theorems in full generality one has to work with the whole log étale site.

4.4.2 Log étale Site For the sake of completeness, let us discuss how one defines the notion of a log étale covering in general. Our motivation is just to see a few more examples from log geometry, and we will not discuss the log étale cohomology theories. Exercise 4.26 Construct an example of surjective log étale morphisms .Y → X and .Z → X such that .Y ×X Z → X is not surjective. In particular, a base change does not have to be surjective, and hence can even be empty. (Hint: for example, one can take the plane with the monoid .N2 , apply log blowing up to the origin and another log blowing up to one of the two preimages of the origin with characteristic .N2 , obtaining a log étale morphism  .X → X with the exceptional divisor consisting of two components .E = E1 ∪ E2 . Then   .Y = X \ E1 and .Z = X \ E2 do the job. It is also instructive to consider a purely combinatorial (or toric) description of this example.) The above example shows that one should be careful with the notion of surjectivity. Naturally, we would like to declare any log blowing up .Y → X to be a cover, but for any .Y   Y the morphism .Y  → X should not be a cover. So, one defines the log étale topology to be the topology generated by Kummer étale covers and log blowings up. Exercise 4.27 Let .f : Y → X be a log étale morphism. Show that f is a log étale cover if and only if for any log blowing up .X → X the morphism .Y ×X X → X is surjective.

5

The Stacks LogS

This section is devoted to a very important technique in logarithmic geometry, which was introduced by Olsson in [14] (with a strong influence of ideas of Luc Illusie). It turns out that S-logarithmic structures on schemes T over (the underlying scheme of) a base log scheme S are classified by a stack .LogS . Working with such stacks allows to interpret various logarithmic constructions and notions in terms of usual algebraic geometry of

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schemes and stacks. In particular, some results can be deduced from the non-logarithmic analogs on the nose.

5.1

Constructions of LogS

5.1.1 The Moduli Definition of LogS To any log scheme .S = (S, MS ) Olsson assigns the category .LogS fibered in groupoids over the category of .S-schemes as follows: an object of .LogS is a logarithmic S-scheme X and a morphism is a strict morphism .Y → X of logarithmic S-schemes. The fiber functor just forgets the log structures. Remark 5.1 (i) If T is an .S-scheme, then an object of .LogS (T ) is just a logarithmic S-scheme whose underlying scheme is T . Thus, the stack .LogS parameterizes the ways in which one can enhance .S-schemes with the structure of logarithmic S-schemes. So, informally speaking, it parameterizes log structures over S. (ii) The association .S → LogS is naturally a functor from the category of schemes to the category of stacks.

5.1.2 Algebraicity Olsson proved that the stack .LogS is in fact an Artin stack of locally finite type over S. The proof goes by checking the usual properties—representability of the diagonal and existence of a smooth presentation. The first property reduces to a simple study of the group of .MS -automorphisms of the log-structures log S-schemes—naturally, they are extensions of diagonalizable groups by finite groups. The second property holds  because . P →Q SP [Q] → LogS is smooth and surjective. The latter will be discussed in Sect. 5.1.5 and then we will use it to construct .LogS very explicitly (in particular, the map from each .SP [Q] factors through the quotient by the group of S-automorphisms, which is the extension of the diagonalizable group .DQgp /P gp by the finite group .AutP (Q)). 5.1.3 The Tautological Log Structure By the definition of the stack .LogS any scheme over it is provided with a canonical log structure, and by descent one immediately obtains that the same is true for stacks over .LogS . In particular, .LogS itself is provided with a tautological log structure .M and for any .x ∈ LogS (T ) the induced homomorphism .(T , Mx ) → (LogS , M) is strict, that is, the log structure .Mx is induced from the tautological structure .M via the structure morphism .T → S. Similarly to Remark 5.1 this provides a formalization of the claim that .M is the universal log structure over S. The following exercise essentially reduces to unravelling the definitions.

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Exercise 5.2 (i) Show that the log structure on S induces a section .S → LogS of the structure morphism .LogS → S. (ii) Let .f : Y → X be an S-morphisms of logarithmic S-schemes and let .x : X → LogS and .y : Y → LogS be the corresponding morphisms. Show that .y = x ◦ f if and only if f is strict. (iii) For a morphism .f : Y → X of log schemes the square

.

is Cartesian if and only if f is strict.

5.1.4 The Stacks XP [Q] The stack .LogX is huge, but it is a rather simple object that can be described by charts very explicitly. A first approximation for this is the following construction due to Olsson. Assume that X is a log scheme with a global chart .X → AP and .P → Q is a homomorphism of monoids. Note that the diagonalizable group .DQgp /P gp = Spec(Z[Qgp /P gp ]) acts on the .AP -scheme .AQ , and hence also acts on the X-scheme .XP [Q] obtained by the base change. Let .XP [Q] denote the quotient stack .[XP [Q]/DQgp /P gp ]. The importance of these stacks introduced by Olsson becomes clear from the universal property they satisfy, which we are going to establish now. A short proof can be found in [8, Lemma 2.2.4]. Exercise 5.3 Let .X → AP be a chart as above and let Y be a logarithmic X-scheme. (i) Show that flat locally on Y the P -homomorphism .Q → MY lifts to a homomorphism .Q → MY and deduce that the functor of such liftings is a .DQgp /P gp -torsor in the flat topology. Moreover, if Y is fs, then this functor is already an étale torsor. (Hint: use gp gp that the homomorphisms .MY  MY has a section flat locally, and this is even true gp étale locally if Y is fs and hence the groups .MY,y are torsion free.) (ii) Deduce that .XP [Q] represents the functor .HomP (Q, MY ) on the category of log schemes over X, while .XP [Q] represents the functor .HomP (Q, MY ). In particular, X-homomorphisms .Y → XP [Q] are in a natural one-to-one correspondence with P homomorphisms .Q → MY . (Hint: the second claim is clear, to deduce the first one divide by the action of .DQgp /P gp = XP [Qgp ] and use (i).) Remark 5.4 Keep the above notation and assume that .Q → MY is a chart that induces an isomorphism .φ : Q = (Y, MY ). There are many liftings of .φ to a chart of Y , obtained

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by multiplying monomials by units, but this is precisely the ambiguity which is killed by dividing by .DQgp /P gp . Thus .Y → XP [Q] can be viewed as the canonical X-chart of Y determined only by the isomorphism .φ. Its only ambiguity is the group .AutP (Q) of P -automorphisms of Q.

5.1.5 A Smooth Presentation Assume that X possesses a global chart .P → MX . Olsson proves that the natural  morphism . P →Q XP [Q] → LogX , where the union is over all homomorphisms from  P to a fine monoid, is strict, surjective and étale. In particular, . P →Q XP [Q] → LogX is a smooth presentation of .LogX . In fact, this easily reduces to the fact that étale locally any logarithmic X-scheme Y possesses an étale cover .Y  → Y whose source possesses a chart .Y  → XP [Q] and hence a strict morphism .Y  → XP [Q]. In general, X possesses a global chart étale-locally, and the above construction is compatible with strict étale morphisms .X → X. So, a presentation of .LogX can be obtained from a presentation of its fine enough strict étale cover. 5.1.6 A Groupoid Presentation  The presentation .f : P →Q XP [Q] → LogX already gives a non-bad approximation of  the source, but clearly it factors through .f  : P →Q XP [Q]/AutP (Q). Even the étale morphism .f  is still not a monomorphism because P -automorphisms of localizations of Q not necessarily come from .AutP (Q), but it is easy to pin down the ambiguity— one needs to identify all localizations of Q in all possible ways, in particular, dividing .XP [Q] by .AutP (Q). Informally speaking, .LogS is obtained from the union of all charts  . P →Q XP [Q] by identifying all isomorphic open subcharts, in particular, dividing by automorphism: at first step this involves dividing by the groups .DQgp /P gp , and then by identifying all localizations of .Qi and .Qj . In a sense, .LogX is nothing else but the universal X-chart constructed purely combinatorially. Now let us outline the construction. It is more convenient to work geometrically, when the contravariant functor .Q → XP [Q] is replaced by the functor .Spec(Q) → XP [Q] from the category of affine Kato fans over .Spec(P ) because the latter globalizes in the obvious way. Moreover, one can naturally define a wider category of Kato stacks, and this functor extends to Kato stacks by (an appropriate) descent. Consider the diagram of all affine P -fans .Spec(Q) with the morphisms being face embeddings, then the colimit .LP exists as a Kato fan. Intuitively, it is a P -“fan” which contains each .Spec(Q) as a face in a unique way. It is not so difficult to show that .LogX = XP [LP ], in particular, the morphism .LogX → X is monoidal (in the stacky sense). Moreover, one can now give an explicit stacky presentation of .LogX . We outline the main results in a (difficult) exercise below and refer to [8, Sections 2,3] for detailed arguments.

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Exercise 5.5 (i) Given P -monoids .Q0 , . . . , Qn by a join face we mean a P -monoid R and face embeddings .Spec(R) → Spec(Qi ) with .0 ≤ i ≤ n (in other words, we fix isomorphisms of R and localizations of .Qi ). Show that the colimit .J (Q0 , . . . , Qn ) of the diagram of all face embeddings of .Q0 , . . . , Qn is a Kato fan, which we call the join of .Q0 , . . . , Qn .  (ii) Construct a natural simplicial Kato fan .Ln = Q0 ,...,Qn J (Q0 , . . . , Qn ) and show that it is in fact a groupoid equivalent to a Kato fan which we denote .LP . In other  words, .L1 = P →Q Spec(Q) → LP is a cover and its fiber powers are .Ln . (iii) Show that .LP is characterized by the following universal property: any P -fan .Spec(Q) possesses a unique face embedding into .L. (iv) Show that .XP [LP ] = LogX and deduce that .LogX is equivalent to the simplicial  stack .Xn = Q0 ,...,Qn XP [J (Q0 , . . . , Qn )], which is, in fact, a groupoid.

5.2

Stacks Log and Logarithmic Properties

Now, we will show how one can systematically interpret various logarithmic properties of morphisms of log schemes. Initially such notions as log smoothness, log flatness, log étaleness, etc. were defined in a rather ad hoc manner. Then in [14] Olsson found a very general way to unify these definitions. Definition 5.6 Let .P be a property of morphisms of schemes, for example, smooth, étale, flat, etc. A morphism of log schemes .f : Y → X is said to be log .P (resp. weakly log .P) if the associated morphism of stacks .Logf : LogY → LogX (resp. .Y → LogX ) is .P. Remark 5.7 (i) Both definitions have some advantages. The morphism .Y → LogX is “smaller” and easier to analyze; if Y is quasi-compact, then a morphism .Y → X factors through an open substack of .LogX finitely presented over X. On the other hand, when studying compositions it is certainly easier to work with the morphisms .LogY → LogX . (ii) Despite the terminology, neither condition implies the other one. Olsson showed in [14, Example 4.3] that if .P is “having geometrically connected fibers”, then log .P does not imply weakly log .P, but there is even a much more basic example: any morphism .f : Y → X with a quasi-compact source is not weakly log surjective because .LogX is never quasi-compact, while f is often log surjective, for example, when it is an isomorphism. However, using that .Y → LogX factors into the composition of an open immersion .Y → LogY and .Logf we obtain that if .P is local on the source, then log .P implies weakly log .P.

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(iii) For the properties of being log smooth, log étale and log flat Olsson showed that the original definitions given by Kato are equivalent to the new ones and also equivalent to the corresponding weak logarithmic properties. Exercise 5.8 Use Olsson’s definition to reprove Kato’s criterion of log smoothness from Theorem 4.12. Most probably, this will lead you to a conceptual explanation of the fact that condition 4.12(ii)(b) on .φ : P → Q is equivalent to smoothness of the morphism .Dk(y),Q → Dk(y),P . Now we can also naturally interpret the notion of a log étale cover: Exercise 5.9 Prove that a morphism .f : Y → X is a log étale cover if and only if Logf : LogY → LogX is an étale cover. In other words, a log étale morphism is a cover if and only if it is log surjective.

.

5.2.1 Equivalence of the Conditions A general result about equivalence of the two definitions was obtained in [8, Theorem 4.3.1]: if .P is stable under pullbacks, étale local on the source, and flat local on the base, then a morphism is log .P if and only if it is weakly log .P. This follows from a slightly surprising fact that .LogY → LogX can be obtained from its small piece .Y → LogX by base change and flat descent. Exercise 5.10 (i) Assume that .Y → X has a global chart .AQ → AP . Show that for any homomorphism .Q → R both squares in the following diagram are Cartesian:

(ii) Using the presentation of the stacks .Log constructed in Sect. 5.1.6, étale descent and claim (i) deduce the assertion of [8, Theorem 4.3.1].

5.2.2 Log Regularity The above equivalence result applies, in particular, to the following properties: smoothness, étaleness, flatness and regularity. In fact, log regularity was introduced in [8] and the equivalence with weak log regularity was used to establish its basic properties, e.g. a chart criterion analogous to Kato’s chart criterion of log smoothness. In fact, this was the original motivation of the research of [8], which, in its turn, was motivated by relative resolution of singularities.

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5.2.3 Logarithmic Differentials The stacks .Log also allow to interpret logarithmic derivations and differentials. In view of the fact that a morphism .f : Y → X is log smooth if and only if the associated morphism 1 1 .h : Y → LogX is smooth, the following fact is very natural: . Y /X = Y /LogX and .DerY /X = DerY /Log (see [2, Lemma 2.4.4]). X Exercise 5.11 (i) Assume that .Y → X possesses a chart .AQ → AP . Prove that .DerY /X = DerY /XP [Q] . (Hint: compare the first fundamental sequences of log derivations associated with .Y → XP [Q] → X and of derivations associated with .Y → XP [Q] → XP [Q].)  (ii) Use the étale cover . Q XP [Q] → LogX to deduce that, indeed, for any morphism of log schemes .Y → X one has that .DerY /X = DerY /LogX and hence also . 1Y /X = 1Y /Log . X

5.2.4 Logarithmic Fibers Let .f : Y → X be a morphism of log schemes. By the log fibers of f we mean the connected components of the fibers of the induced morphism .h : Y → LogX . For example, if f is log smooth or log regular, then the log fibers are smooth or regular, respectively. It is easy to compute the log fibers étale-locally: if f has a chart modeled on .AQ → AP , then h factors through the étale morphism .XP [Q] → LogX hence the log fibers are nothing else but the fibers of the stacky chart .Y → XP [Q]. Let us consider two general types of examples of opposite kind. If the homomorphisms .Mx → My are injective, then the log fibers have the most natural geometric interpretation: Exercise 5.12 Assume that .f : Y → X has a chart modeled on .AQ → AP with sharp P and Q (such a chart exists étale-locally at .y ∈ Y with .x = f (y) if .Mx → My is injective). Show that the log fibers of f are the connected components of the log strata of the fibers of f . In particular, log fibers of a toroidal variety over a field are just the connected components of its log stratification. Such a description certainly cannot work for log blowings up, which might have nondiscrete fibers but are log étale. Exercise 5.13 Show that the log fibers of any log blowing up .Y → X are nothing else but the points of Y (as one would expect in the case of a monomorphism). In general, one can locally factor a morphism into a composition of a sharp morphism and a log blowing up, so log fibers admit a sort of a mixed description, but we do not discuss this here and refer the interested reader to [2, §2.2].

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Acknowledgments I am very grateful to Luc Illusie for telling the story of discovery of log schemes and for reading the notes and making many helpful comments.

References 1. D. Abramovich, Q. Chen, D. Gillam, Y. Huang, M. Olsson, M. Satriano, S. Sun, Logarithmic Geometry and Moduli, Handbook of Moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24 (Int. Press, Somerville, MA, 2013), pp. 1–61. MR 3184161 2. D. Abramovich, M. Temkin, J. Włodarczyk, Relative desingularization and principalization of ideals, arXiv e-prints (2020), arXiv:2003.03659 3. W. Fulton, Intersection Theory, 2nd edn., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2 (Springer, Berlin, 1998). MR 1644323 4. L. Illusie, From Pierre Deligne’s secret garden: looking back at some of his letters. Jpn. J. Math. 10(2), 237–248 (2015). MR 3392531 5. K. Kato, Logarithmic Structures of Fontaine-Illusie, Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988) (Johns Hopkins Univ. Press, Baltimore, MD, 1989), pp. 191–224. MR 1463703 (99b:14020) 6. K. Kato, Toric singularities. Amer. J. Math. 116(5), 1073–1099 (1994). MR 1296725 (95g:14056) 7. G. Kempf, F.F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973). MR 0335518 (49 #299) 8. S. Molcho, M. Temkin, Logarithmically regular morphisms. Math. Ann. 379(1–2), 325–346 (2021). MR 4211089 9. C. Nakayama, Logarithmic étale cohomology. Math. Ann. 308(3), 365–404 (1997). MR 1457738 10. C. Nakayama, Logarithmic étale cohomology, II. Adv. Math. 314, 663–725 (2017). MR 3658728 11. W. Nizioł, Toric singularities: log-blow-ups and global resolutions. J. Algebraic Geom. 15(1), 1–29 (2006). MR 2177194 (2006i:14015) 12. W. Nizioł, K-theory of log-schemes. I. Doc. Math. 13, 505–551 (2008). MR 2452875 13. A. Ogus, Lectures on Logarithmic Algebraic Geometry, Cambridge Stud. Adv. Math., vol. 178 (Cambridge University Press, Cambridge, 2018), xviii+539 pp. ISBN:978-1-107- 18773-3 14. M.C. Olsson, Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36(5), 747–791 (2003). MR 2032986 15. J. Włodarczyk, Functorial resolution except for toroidal locus. Toroidal compactification. Adv. Math. 407, Paper No. 108551, 103 pp. (2022)

Birational Geometry Using Weighted Blowing Up Dan Abramovich, Michael Temkin, and Jarosław Włodarczyk

1

Introduction

1.1

The Place of Resolution

Resolution of singularities, when available, is one of the most powerful tool at the hands of an algebraic geometer. One would wish it to have a completely intuitive, natural proof. Many such proofs exist for curves, see [21, Chapter 1]. At least one completely conceptual proof exists for surfaces in characteristic .>2 and for threefold of characteristic .>5, see [11]. It is fair to say that with most other cases, including Hironaka’s monumental proof of resolution of singularities in characteristic zero [16], one feels that varieties resist our

The research presented here is supported by BSF grant 2014365, ERC Consolidator Grant 770922— BirNonArchGeom, and NSF grant DMS-1759514. D. Abramovich () Department of Mathematics, Brown University, Providence, RI, USA e-mail: [email protected] M. Temkin Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel e-mail: [email protected] J. Włodarczyk Department of Mathematics, Purdue University, West Lafayette, IN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_4

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resolution efforts, and one has to resort to extreme measure to force resolution upon them. On the other hand, one feels that it should not be so: myriad applications show that it is in the best interest of any variety to be resolved.

1.2

This Note

The purpose of this note is to recall how this “resistance” comes about, and to outline a new approach, where we are able to prove resolution of singularities in characteristic 0 with the complete cooperation of our varieties, by infusing the theory with a bit of modern moduli theory, specifically the theory of algebraic stacks. The main result, explained throughout this note, is the following: Theorem 1.2.1 (Weighted Hironaka) There is a procedure F associating to a singular subvariety .X ⊂ Y embedded with pure codimension c in a smooth variety Y over a field of characteristic 0, a center .J¯ with blowing up .Y  → Y and proper transform .(X ⊂ Y  ) = F (X ⊂ Y ) such that .maxinv(X ) < maxinv(X). In particular, for some n the iterate ◦n (X ⊂ Y ) of F has .X smooth. .(Xn ⊂ Yn ) := F n For this purpose we systematically use stack-theoretic weighted blowings up. As is well known, the use of classical blowings up of smooth centers has been of great value in birational geometry. We end this note by discussing to what extent stack-theoretic weighted blowings up can be as useful. Theorem 1.2.1 is our main theorem in [2]. An almost identical result is given concurrently in [22]. The methods employed are quite different.

2

Curves and Surfaces

Our aim is embedded resolution, namely to resolve a singular subvariety .X ⊂ Y of a smooth variety Y by applying birational transformations on Y .

2.1

How to Resolve a Curve?

To resolve a singular curve C (1) find a singular point .x ∈ C, and then (2) blow it up.

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This procedure always works: Fact (See [15, Theorem V.3.8]) .pa (C) gets smaller in each such step, hence the procedure ends with a smooth curve.

2.2

How to Resolve a Surface?

Surfaces are more complex, as their singularities can reside on either curves or points. To resolve a singular surface S one wants to (1) find the worst singular locus .C ⊂ S, and then (2) show that C is smooth, and blow it up. However: Fact This in general does not get better.

2.3

Example: Whitney’s Umbrella

Consider .S = V (x 2 − y 2 z). The worst singularity is the origin. In the z chart we get 2 2 2 3 .x = x3 z, y = y3 z, giving .x z − y z = 0, or 3 3 z2 (x32 − y32 z) = 0.

.

The first term is exceptional, which we may ignore. However the second is the same as S. It appears that we gained nothing.

2.4

How to Resolve a Surface: Classical Approach

Of course surface resolution can be achieved. The standard algorithms in characteristic 0— which applies in arbitrary dimension—calls for recording the exceptional divisors. Thus after the first blowing up the equation .x32 − y32 z has a distinguished coordinate z, which should be thought of as an improvement. But even surfaces do not like this approach and resist it kicking and screaming: the exceptional divisors get in the way of the standard natural algorithm, which uses hypersurfaces of maximal contact recalled later on. In short, such hypersurfaces are not necessarily transverse to the exceptional divisors, giving no end of trouble.

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By what seems like pure luck, one can introduce an auxiliary subroutine of resolution to throw exceptional divisors out of the picture. In terms of singularity invariants, it is not enough to record the exceptional locus, but also some of its history—or the “state of the algorithm”—is needed. Of course this is all counterintuitive—we introduce these divisors to make progress, and yet we introduce a procedure to get them out of the way—but somehow this nevertheless works.

3

Explaining the Main Result

Coming back to Theorem 1.2.1, our first goal is to explain this result and how it could possibly avoid the complications discussed in the previous section.

3.1

Functoriality

First things first: here procedure means a functor for smooth surjective morphisms: if f : Y1  Y is smooth then .J1 = f −1 J and .Y1 = Y1 ×Y Y  , and .X can be taken to be the proper transform (in the course of the proof we actually use the so called weak transform instead). The final result .(Xn ⊂ Yn ) is actually functorial for smooth but not necessarily surjective morphisms—only the number n is not. Functoriality has great value beyond elegance—it guarantees that resolution is equivariant under automorphisms, it is compatible with localization, and in particular smooth points are not disturbed. Functoriality is present in Hironaka’s later work (under the term “canonical resolution”), and is clarified in the works of Villamayor [25] and Bierstone–Milman [5, 6]. It is stated explicitly in Andrew J. Schwartz’s thesis [24]. Włodarczyk [27] was the first to show that functoriality is a powerful tool for the proof itself, a method we employ here. Indeed, the result of our blowing up is an algebraic stack, but functoriality allows us to replace it by a presentation by schemes, so we can have the input of the theorem be a scheme. We proceed with an overview of the concepts we use, exhibiting them in examples, and then defining them more thoroughly later on.

.

3.2

Preview on Invariants

We use singularity invariants to guide the procedure. For .p ∈ X we define invp (X) ∈  ⊂

.

and show

Q≤n ≥0 ,

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Theorem 3.2.1 ([2, Theorem 5.2.1]) (1) . is a well-ordered subset with respect to the lexicographical ordering, (2) .invp (X) is lexicographically upper-semi-continuous, and (3) .p ∈ X is smooth if and only if .invp (X) = min . We define .maxinv(X) = maxp invp (X). Example 3.2.2 .invp (V (x 2 − y 2 z)) = (2, 3, 3) The idea is: the variable x appears in the equation .x 2 − y 2 z in the monomial of lowest possible degree 2, and the variables y and z appear in a monomial of the next lowest degree 3. Remark 3.2.3 These invariants have been in our arsenal for ages. All the great works of the last three decades on resolution in characteristic 0 use this invariant, with additional information interspersed within it.

3.3

Preview of Centers

We use centers denoted J for comparing singularities and the notion of admissibility, and associated reduced centers denoted .J¯ for blowing up. As in the example above, invariants are determined using associated local parameters. If .invp (X) = maxinv(X) = (a1 , . . . , ak ) then, locally at p, we have J = (x1a1 , . . . , xkak ).

.

Now we normalize the center by a rescaling procedure: write (a1 , . . . , ak ) = (1/w1 , . . . , 1/wk )

.

with .wi ,  ∈ N and .gcd(w1 , . . . , wk ) = 1. We set the reduced center to be 1/w 1/w J¯ = (x1 1 , . . . , xk k ).

.

Defining the center J requires a new formalism: first, we will see that the parameters .xi require a choice, while we claim below the center is uniquely defined. Second, the center is a beast involving fractional powers of parameters, something that goes beyond the familiar world of ideals. We will explain these points later in this note.

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Example 3.3.1 For .X (x 1/3 , y 1/2 , z1/2 ).

= V (x 2 − y 2 z) we have .J

=

(x 2 , y 3 , z3 ) and .J¯

=

This example is a bit too easy, since the variables .x, y and z were staring us in the face, as the ideal is binomial. We will revisit this issue below where a different example will be the guide. Remark 3.3.2 The center J has been staring in our face for a while. If one interprets J in terms of Newton polyhedra, it appears in section 1 of Youssin’s thesis [28]. Youssin’s construction is a simplified variant of Hironaka’s characteristic polyhedron of a singularity, see [17].

3.4

Example: Blowing Up Whitney’s Umbrella x 2 = y 2 z

The blowing up .Y  → Y makes .J¯ = (x 1/3 , y 1/2 , z1/2 ) principal. Explicitly: the third chart (corresponding to z) has .x = w 3 x3 , y = w 2 y3 , z = w 2 with chart Y  = [ Spec C[x3 , y3 , w] / (±1) ],

.

on which .(±1) acts by .(x3 , y3 , w) → (−x3 , y3 , −w). The transformed equation is w 6 (x32 − y32 ),

.

and the invariant of the proper transform .(x32 − y32 ) is .(2, 2) < (2, 3, 3). Remark 3.4.1 In fact, people studying explicit birational geometry, as well as people studying explicit moduli spaces of surfaces, have known all along that X prefers to be blown up in this way.

3.5

Definition of the Weighted Blowing Up Y  → Y

We are now ready to define our blowing up in general. 1/w 1/w Let .J¯ = (x1 1 , . . . , xk k ). Define the graded algebra AJ¯

.

⊂

OY [T ]

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as the integral closure of the image of

.

Let S0 ⊂ SpecY AJ¯ ,

.

S0 = V ((AJ¯ )>0 )

be the vertex of the spectrum. Then BlJ¯ (Y ) := Proj Y AJ¯ :=

.

   (Spec AJ¯  S0 ) Gm .

This is the analogue of the definition of usual blowing up one can find in [15, Example 7.12.1], where .Yi is sent to .xi T , namely placed in degree 1, whereas here it is placed in degree .wi . Unlike Hartshorne’s description, we take the stack theoretic quotient rather than the scheme theoretic quotient. This is critical if one is to have a smooth ambient space after blowing up.

3.6

Description of Y  → Y

Just like regular blowing up, a weighted blowing up has a local description in terms of charts. The .x1 -chart is [Spec k[u, x2 , . . . , xn ] / μw1 ],

.

with .x1 = uw1 and .xi = uwi xi for .2 ≤ i ≤ k, and induced action:  (u, x2 , . . . , xn ) → (ζ u , ζ −w2 x2 , . . . , ζ −wk xk , xk+1 , . . . , xn ).

.

For our discussion at the end of this paper it is also useful to have a local toric description of the blowing up. We follow [10]. For simplicity let us assume Y is affine space, corresponding to the cone .σ = Rn≥0 with lattice .Nn . Then .Y  is the toric stack corresponding to the star subdivision . := vJ¯  σ along vJ¯ = (w1 , . . . , wk , 0, . . . , 0),

.

with the cone σi = vJ¯ , e1 , . . . , eˆi , . . . , en 

.

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endowed with the sublattice .Ni ⊂ N generated by the elements vJ¯ , e1 , . . . , eˆi , . . . , en ,

.

for all .i = 1, . . . , k. The coarse moduli space of .Y  is simply the toric variety corresponding to the star subdivision, without the auxiliary sublattices.

3.7

Determining the Center J Associated to an Ideal I in Examples

3.7.1 An Example with Fractional Powers Consider .X = V (x 5 + x 3 y 3 + y 8 ) at .p = (0, 0); write .I := IX . Define .a1 = ordp I = 5, and choose .x1 to be any variable appearing in a degree-.a1 term, for instance x. This determines the beginning of our center .JI = (x 5 , y  ). We note that if we were to change variables, we could use .x + y 2 instead, so there is definitely a choice involved. To balance .x 5 with .x 3 y 3 we need .x 2 and .y 3 to have the same weight, implying that .x 5 and .y 15/2 have the same weight. If we were to balance with the term .y 8 we would have taken .y 8 instead. Since .15/2 < 8 the choice .y 15/2 dominates, and we use JI = (x 5 , y 15/2 )

and

.

J¯I = (x 1/3 , y 1/2 ).

3.7.2 A Related Example If instead we took .X = V (x 5 + x 3 y 3 + y 7 ), then since .7 < 15/2 we would use JI = (x 5 , y 7 )

and

.

3.8

J¯I = (x 1/7 , y 1/5 ).

Describing the Blowing Up in the New Examples

(1) Considering X = V (x 5 + x 3 y 3 + y 8 ) at p = (0, 0), • the x-chart has x = u3 , y = u2 y1 with μ3 -action, and equation u15 (1 + y13 + uy18 )

.

with smooth proper transform. • The y-chart has y = v 2 , x = v 3 x1 with μ2 -action, and equation v 15 (x15 + x13 + u)

.

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with smooth proper transform. (2) Considering X = V (x 5 + x 3 y 3 + y 7 ) at p = (0, 0), • the x-chart has x = u7 , y = u5 y1 with μ7 -action, and equation u35 (1 + uy13 + y17 )

.

with smooth proper transform. • The y-chart has y = v 5 , x = v 7 x1 with μ5 -action, and equation v 35 (x15 + ux13 + 1)

.

with smooth proper transform.

3.9

Coefficient Ideals

We need a mechanism for induction on dimension. The first example shows clearly that one can’t just restrict the ideal .I to .{x1 = 0}, since this loses information of monomials mixing .x1 and other variables. These mixed monomials are revealed by taking derivatives of the ideal .I. We thus must restrict to .x1 = 0 the data of all I, DI, . . . , Da1 −1 I

.

with corresponding weights a1 , a1 − 1, . . . , 1.

.

We combine these in C(I, a1 ) :=



.

  f I, DI, . . . , Da1 −1 I , ba

−1

1 where f runs over monomials .f = t0b0 · · · ta1 −1 with weights

 .

bi (a1 − i) ≥ a1 !.

We now define the restricted coefficient ideal .I[2] = C(I, a1 )|x1 =0 .

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We note however that .I[2] naturally has weight .a1 !, whereas .I has weight .a1 . We need to compensate for this by “rescaling” .I[2] down to degree .a1 . This is the source of fractional invariants and fractional powers in our centers. The coefficient ideal we use here was introduced in [21, §3.54]. It is a variant of constructions appearing in [5, 25, 27].

3.10

Defining JI in General

Definition 3.10.1 Let a1 = ordp I, with x1 a regular element in Da1 −1 I—a maximal contact element. Suppose I[2] has invariant invp (I[2]) defined with parameters x¯2 , . . . , x¯k on {x1 = 0}, with lifts x2 , . . . , xk in OY . Set  invp (I[2]) .invp (I) = (a1 , . . . , ak ) := a1 , (a1 − 1)! and JI = (x1a1 , . . . , xkak ).

.

Example 3.10.2 (1) for X = V (x 5 + x 3 y 3 + y 8 ) we have I[2] = (y)180 , so JI = (x 5 , y 180/24 ) = (x 5 , y 15/2 ).

.

(2) for X = V (x 5 + x 3 y 3 + y 7 ) we have I[2] = (y)7·24 , so JI = (x 5 , y 7 ).

3.11

What Is J ?

We have postponed the question—what kind of beast is J ? It needs to allow for rational powers, and different choices of parameters giving rise to .J1 = (x 5/2 , y 5/2 ) and .J2 = ((x + y)5/2 , (x − 5y)5/2 )) must have .J1 = J2 . We follow ideas permeating birational geometry to resolve this issue. Definition 3.11.1 Consider the Zariski-Riemann space .ZR(X) with its sheaf of ordered groups ., and associated sheaf of rational ordered group . ⊗ Q. • A valuative .Q-ideal is

 γ ∈ H 0 ZR(X), ( ⊗ Q)≥0 ) .

.

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• Each valuative .Q-ideal induces a “shadow” ideal on Y by .Iγ := {f ∈ OX : v(f ) ≥ γv ∀v}. • Conversely, any coherent ideal .I induces a valuative .Q-ideal denoted .v(I) := (min v(f ) : f ∈ I)v . A center is in particular a valuative .Q-ideal.

4

Elements of the Proof

Now that we have defined our terms, we briefly describe key elements of the proof.

4.1

Homogeneity

Let .I ⊂ OY and assume .x1 ∈ D≤a−1 I is a maximal contact element at .p ∈ Y . The ideals .C(I, a1 ) is MC-invariant in the sense of [21, §3.53], hence it is homogeneous in the sense of [27]: Theorem 4.1.1 ([27, Lemma 3.5.5], [21, Theorem 3.92]) Let .x1 , x1 be maximal contact elements at p, and .x2 , . . . , xn ∈ OY,p such that .(x1 , x2 , . . . , xn ) and .(x1 , x2 , . . . , xn ) are both regular sequences. There is a scheme .Y˜ with point .p˜ ∈ Y˜ and two morphisms  ˜ → Y with .φ(p) .φ, φ : Y ˜ = φ  (p) ˜ = p, both étale at p, satisfying (1) .φ ∗ x1 = φ  ∗ x1 , (2) .φ ∗ xi = φ  ∗ xi for .i = 2, . . . , n, and (3) .φ ∗ C(I, a1 ) = φ  ∗ C(I, a1 ). This in particular implies that replacing .x1 by .x1 while keeping the other parameters intact preserves the whole procedure.

4.2

Formal Decomposition

A useful close cousin of homogeneity is a convenient formal decomposition of coefficient ideals, obtained by diagonalizing logarithmic differential operators: Lemma 4.2.1 ([2, Lemma 4.4.1]) If .ordp (I) = a1 and .x1 a corresponding maximal contact, then in .Cx1 , . . . , xn  we have C(I, a) = (x1a1 ! ) + (x1a1 !−1 C˜1 ) + · · · + (x1 C˜a1 !−1 ) + C˜a1 ! ,

.

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where Ca1 ! ⊂ (x2 , . . . , xn )a! ⊂ kx2 , . . . , xn ,

.

where .Cj −1 := D≤1 (Cj ) satisfy .Ck Cl ⊂ Ck+l , and .C˜j = Cj kx1 , . . . , xn .

4.3

Admissibility and Coefficient Ideals

Admissibility of centers is a notion used throughout resolution of singularities. The key point is that if one blows up an admissible center, some measurement of the singularity does not get worse. Definition 4.3.1 J is .I-admissible if .v(J ) ≤ v(I). Lemma 4.3.2 ([2, Section 5.3.1]) This is equivalent to .IOY  = E  I  , with .E ⊂ OY  the ideal of the exceptional divisor, .J = J¯ and .I  an ideal. Indeed, on .Y  the center J becomes .E  , in particular principal. Proposition 4.3.3 ([2, Lemma 5.3.7]) A center J is .I-admissible if and only if .J (a1 −1)! is .C(I, a1 )- admissible. This is a consequence of formal decomposition.

4.4

The Key Theorems

It is now clear what remains to prove: Theorem 4.4.1 ([2, Theorem 5.2.1 and 5.6.1]) The center .JI is well-defined and functorial. Theorem 4.4.2 ([2, Theorem 5.4.1]) The center .JI is .I-admissible. Theorem 4.4.3 ([2, Theorem 5.5.1]) 

(1) .C(I, a1 )OY  = E  C  with .invp C  < invp (C(I, a1 )). (2) .IOY  = E  I  with .invp I  < invp (I). Remarkably, each of these theorems follows with little effort from homogeneity, formal decomposition, and induction.

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Birational Geometry and Blowing Up

5.1

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Smooth blowings up are remarkably useful in birational geometry in characteristic 0. One can point out to two salient features making them useful: (1) One can describe the change of geometry resulting from a smooth blowing up .Y  = BlZ (Y ) of a smooth variety Y , describing the geometry of .Y  in terms of that of Y , Z, and the position of Z in Y . (2) If any two smooth varieties .Y1 , Y2 are related by a proper birational map, they are in fact connected through smooth blowings up. Item (2) is Weak Factorization, see [1, 26]. It directly implies structural result, such as Bittner’s theorem on the Grothendieck ring of varieties [7, Theorem 3.1]. Examples of item (1) include the formula [14, Proposition 6.7 and Example 8.3.9] for the intersection theory of .Y  , and Bondal and Orlov’s semiorthogonal decomposition of .D(Y  ) [8, Theorem 4.2]. Together (1) and (2) imply results on birational invariance of certain biregular invariants, for instance Borisov and Libgober’s results on elliptic genera [9].

5.2

Birational Geometry Using Weighted Blowing Up

The situation of weighted blowings up is similar but incomplete: (1) There is no doubt one can describe the change of geometry resulting from a weighted blowing up .Y  = BlJ (Y ). (2) It follows from Weak Factorization and Bergh’s destackification [4] that if two smooth stacks .Y1 , Y2 are related by a proper birational map, they are in fact connected through weighted blowings up. Regarding (2), indeed the paper [27] preceding weak factorization provides a factorization in weighted blowings up and down for smooth varieties. Hu [18] showed how to directly extend such factorization results to varieties with orbifold singularities. As an example for (1), Kawamata [20, Section 5] generalized the theorem of Bondal and Orlov in great generality. Inchiostro [19] described the Deligne–Mumford-Knudsen moduli stack .M1,2 of stable, 2-pointed curves of genus 1, as a stack-theoretic weighted blowup of the pseudostable space, which explicitly is the weighted projective stack .P(2, 3, 4). From this Inchiostro deduced the Brauer group and Chow ring of this stack. A general formula for the integral Chow ring of a weighted blowup is given in the forthcoming [3]. It appears that the only reason other aspects are not readily available is that people did not have weighted blowings up in mind!

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One point of caution though: weighted blowings up exhibit codimension-1 phenomena, √ as they include root constructions .Y  = Y ( r D) along smooth divisors D. For instance, the plurigenera of .Y  are in general bigger than those of Y .

5.3

Strong Factorization of Toric Maps

Oda [23] conjectured that if .Y, Y  are smooth toric varieties related by a proper toric birational map, then there is a sequence of smooth toric blowings up .Y   Y such that .Y   Y  is also a morphism factoring as a sequence of smooth blowings up. This is still a conjecture even in dimension 3. The paper [12] describes an algorithm which, if it terminates, provides such a factorization in general. The algorithm was shown to terminate on millions of cases in dimension 3. The same algorithm should apply for birational toric stacks and weighted blowings up, a more general, and therefore harder, case. However Ewald [13] showed that any two three-dimensional fans .1 , 2 with the same support are related via a sequence of star subdivisions, if these are allowed to be centered at points which are not unimodular barycenters. This immediately implies the following: Corollary 5.3.1 (The Weighted Strong Oda’s Conjecture for Threefold) Let .Y, Y  be smooth three dimensional toric stacks related by a proper toric birational map. Then there is a sequence of weighted blowings up .Y   Y such that .Y   Y  is also a morphism factoring as a sequence of weighted blowings up. Ewald’s factorization algorithm is a greedy algorithm using the three-dimensional situation. It is not known if it can be generalized to higher dimensions: Conjecture 5.3.2 (The Weighted Strong Oda’s Conjecture) Let .Y, Y  be smooth toric stacks of dimension .> 3 related by a proper toric birational map. Then there is a sequence of weighted blowings up .Y   Y such that .Y   Y  is also a morphism factoring as a sequence of weighted blowings up. This is evidently weaker than the usual strong Oda’s conjecture. Given that Oda’s conjecture is unsolved in decades, one might give this possibly easier question a try. Note that even here weighted blowings up result in codimension-1 phenomena through the back door: consider the cone .σ = R≥0 with the lattice .N2 . Let .1 be the standard subdivision along .(1, 1), and let .2 be the stacky fan obtained by subdividing along .(1, 2), with lattices generated by edge generators. The stellar subdivision .1 of .1 along .(1, 2) is not the same as the stellar subdivision .2 of .2 along the ray of .(1, 1), since this ray is generated by .(2, 2) in .2 ! Rather .2 → 1 is the lattice alteration of the corresponding root construction.

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References 1. D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk, Torification and factorization of birational maps. J. Amer. Math. Soc. 15(3), 531–572 (2002) (electronic). MR 1896232 (2003c:14016) 2. D. Abramovich, M. Temkin, J. Włodarczyk, Functorial embedded resolution via weighted blowings up, arXiv e-prints (2019), arXiv:1906.07106 3. V. Arena, S. Obinna, The integral Chow ring of a weighted blowup, arXiv e-prints (2023), arXiv:2307.01459 4. D. Bergh, Functorial destackification of tame stacks with abelian stabilisers. Compos. Math. 153(6), 1257–1315 (2017) 5. E. Bierstone, P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997). MR 1440306 (98e:14010) 6. E. Bierstone, P.D. Milman, Functoriality in resolution of singularities. Publ. Res. Inst. Math. Sci. 44(2), 609–639 (2008). MR 2426359 7. F. Bittner, The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004). MR 2059227 8. A. Bondal, D. Orlov, Derived categories of coherent sheaves, in Proceedings of the International Congress of Mathematicians, vol. II (Beijing, 2002) (Higher Ed. Press, Beijing, 2002), pp. 47– 56. MR 1957019 9. L. Borisov, A. Libgober, McKay correspondence for elliptic genera. Ann. Math. (2) 161(3), 1521–1569 (2005). MR 2180406 (2008b:58030) 10. L.A. Borisov, L. Chen, G.G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc. 18(1), 193–215 (2005). MR 2114820 11. S.D. Cutkosky, Resolution of singularities for 3-folds in positive characteristic. Amer. J. Math. 131(1), 59–127 (2009). MR 2488485 12. S. Da Silva, K. Karu, On Oda’s strong factorization conjecture. Tohoku Math. J. (2) 63(2), 163– 182 (2011). MR 2812450 13. G. Ewald, Über stellare Unterteilung von Simplizialkomplexen. Arch. Math. (Basel) 46(2), 153– 158 (1986). MR 834828 14. W. Fulton, Intersection Theory, 2nd edn., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2 (Springer, Berlin, 1998). MR 1644323 15. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52 (Springer, New York, 1977). MR 0463157 (57 #3116) 16. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964); ibid. (2) 79, 205–326 (1964). MR 0199184 17. H. Hironaka, Characteristic polyhedra of singularities. J. Math. Kyoto Univ. 7, 251–293 (1967). MR 0225779 18. Y. Hu, Factorization theorem for projective varieties with finite quotient singularities. J. Differential Geom. 68(3), 545–551 (2004). MR 2144541 19. G. Inchiostro, Moduli of genus one curves with two marked points as a weighted blow-up. Math. Z. 302(3), 1905–1925 (2022). MR 4492520 20. Y. Kawamata, Derived categories of toric varieties. Michigan Math. J. 54(3), 517–535 (2006). MR 2280493 21. J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, NJ, 2007). MR 2289519 (2008f:14026)

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22. M. McQuillan, Very functorial, very fast, and very easy resolution of singularities. Geom. Funct. Anal. 30(3), 858–909 (2020). MR 4135673 23. T. Oda, Torus Embeddings and Applications, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 57 (Tata Institute of Fundamental Research/Springer, Bombay/Berlin, 1978). Based on joint work with Katsuya Miyake. MR 546291 24. A.J. Schwartz, Functorial smoothing of morphisms in equal characteristic 0, Ph.D. thesis, Harvard University, 1992, p. 68. MR 2687722 25. O. Villamayor, Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989). MR 985852 26. J. Włodarczyk, Toroidal varieties and the weak factorization theorem. Invent. Math. 154(2), 223– 331 (2003). MR 2013783 (2004m:14113) 27. J. Włodarczyk, Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc. 18(4), 779–822 (2005) (electronic). MR 2163383 28. B. Youssin, Newton polyhedra without coordinates. Mem. Amer. Math. Soc. 87(433), i–vi, 1–74 (1990). MR 1013093

Relative and Logarithmic Resolution of Singularities Michael Temkin

1

Introduction

When working on this chapter my original plan was to provide an expanded version of lecture notes of a mini-course on logarithmic resolution and semistable reduction, but while working on this I decided to widen the perspective and discuss the non-logarithmic methods too—both the classical method and the weighted (or dream) algorithm. This produces a certain intersection with the material of the other chapters, but I think that the profit is larger than this inconvenience. So, what does one gain? Primarily, when working on the exposition of the logarithmic methods I found a new perspective, which allows to relatively uniformly present and compare all four methods—the classical one, the logarithmic one, the non-logarithmic weighted method and the logarithmic weighted method. This perspective is slightly new even in the case of the classical algorithm— marked ideals .(I, d) are not used, but we blow up d-multiple smooth centers instead—so we will start with a reinterpretation of the classical method (assuming a basic familiarity with it), and then add a logarithmic layer, a stack-theoretic layer, and a weighted layer, each time either obtaining a new algorithm or studying where an attempt fails and preparing

This research is supported by BSF grants 2014365 and 2018193, ERC Consolidator Grant 770922— BirNonArchGeom. M. Temkin () Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_5

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to add one more layer to fix this failure. Not only does this presentation follow the order of discovery of these methods and layers, but I hope it also makes the exposition more accessible, since we try to introduce new tools one after another rather than all at once. Second, in any case the weighted logarithmic method of Quek should be covered by this chapter, so it makes sense to start with the non-logarithmic dream algorithm, and only then add the logarithmic layer, indicating the needed logarithmic adjustments. Our goal is to introduce all relevant notions, constructions and techniques, and to formulate all main results, including all important intermediate results. There are no proofs in the notes. Some easier results are given as exercises and provided with hints, more difficult theorems are provided with references to the literature and a short discussion of main ideas of the proof. So these notes can be viewed as a light guide or a companion for reading research papers, where the new methods were constructed: [5, 7, 8] (see also [25], though it uses a different language) and [30].

1.1

History and Motivation

Until a few years ago there was known an essentially unique basic functorial method for principalization of ideals and resolution of varieties in characteristic zero, to which we refer in the sequel as “the classical method”. It was distilled during decades from Hironaka’s original method from [19], and in this joint and very long effort took part Hironaka himself (idealistic exponents, see [20]), Giraud (maximal contact, see [17]), Villamayor and Bierstone-Milman (canonicity, see [38] and [11]), Schwartz (smooth functoriality of Hironaka’s resolution, see [31]), Włodarczyk (construction that uses smooth functoriality), and others. In particular, until 2017 it was not clear if there exist other algorithms, especially the ones which are simpler, faster or possess better functorial properties. These questions, fundamental by themselves, are especially important in view of the fact that a similar method fails in two other classical desingularization problems: resolution in positive characteristic and resolution of vector fields (in characteristic zero). So, enriching the pool of ideas and techniques can be critical in order to achieve a substantial progress in these questions. The first advance beyond the classical settings was done when a logarithmic analogue of the classical method was constructed for varieties in [7] and then extended to morphisms (or semistable reduction theorems) in [8]. The original motivation for this project was twofold: (1) we wanted to obtain a functorial semistable reduction theorem, which will extend, in particular, to any valuation ring, not necessarily discrete, (2) we wanted to clarify the role of log structures in the classical algorithm, where it was visible but a bit opaque. Both lines pointed in the same direction: (1) Over a general valuation ring the best one can hope for is the semistable reduction in the sense of Abramovich-Karu (see [2] and [9]), and this indicates that one has to work with arbitrary fs log structures and not only those with free monoids .M x . In addition,

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the notions of smoothness, derivations, smooth functoriality, etc. should be replaced by log smoothness, log derivations, log smooth functoriality, etc. (2) In this context it was natural to seek for principalization on general log smooth varieties as opposed to smooth varieties with an snc boundary (or the induced log structure). All in all, one is led to replace the classical setup by the logarithmic one, and the rest, to our surprise, was rather imposed upon us (though not easy to discover). In particular, it turned out that the only sufficiently general class of logarithmic centers that preserve log smoothness are intersections of subvarieties with monomial centers. A relevant log algorithm is even simpler than the classical one because no induction on the log stratification is needed anymore, but it got stuck at one place and insisted that we also blow up centers generated by roots of monomials. Such a blowing up may be not log smooth, and to resolve this we had to refine such blowings up to stacks—adding the stacktheoretic layer is the solution which seemed to us technical but unavoidable. In fact, we just introduced a stack-theoretic refinement of the classical weighted blowing up (with a certain pattern of weights). Note that analogous obstacles were discovered earlier in resolution of vector fields, and the solution also was to consider non-representable weighted blowings up (or any equivalent tool playing the same role), see [28] and [26]. Once a new pool of blowings up that preserve smoothness (in the setting to stacks) was discovered, the next natural question was to study which improvement to the classical algorithm can be obtained using them all. It was independently studied in [25] (following [26]) and in [5] (following [7]) and, as in the classical case, led to the same algorithm, despite different description and justification. Quite to our surprise, the natural principalization algorithm in this setting does not involve any log structure, does not have any memory, uses a simple (in fact, classical) multi-order invariant and improves it after each single weighted blowing up. Moreover, even the resulting resolution algorithm has the same properties, and it is really a non-embedded method. Finally, such an algorithm was believed not to exist (and it does not exist in the classical setting, see Sect. 6.4.5). For all these reasons we sometimes call it a dream algorithm, despite the fact that (unfortunately for us) some experts consider it as a variation of old ideas, which does not contain anything essentially new. . . Finally, one may ask if a dream algorithm also exists in the logarithmic situation, in particular, leading to a fast and simple resolution of schemes with divisors and semistable reduction theorem. The answer is yes. The absolute case was worked out in [30], and it seems certain that the similar method will also apply in the relative setting.

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An Overview

Now let us outline the content of these notes.

1.2.1 The General Principles In Sect. 2 we formulate the general principles that apply to all four methods known so far. In fact, this is an excellent time for such a classification—we already have a few known functorial methods (unlike the past decades, when only the essentially unique classical method was known), but still quite a few methods, so that such a generalization is possible. . . In brief, quite surprisingly each algorithm seems to be quite determined by what we call the framework of the method: the class of geometric objects one works with, the relevant notions of smoothness and derivations, the admissible centers one can blow up without destroying the smoothness, and a primary classification of admissible centers by a (partially or totally) ordered set whose elements are called orders, log orders, weighted orders, etc. Resolution of Z is always deduced from principalization of .IZ ⊂ OX on a manifold X in which Z is (locally) embedded, and the principalization of .I ⊆ OX is deduced from appropriate order reduction of .I, in which one iteratively blows up an admissible center .J which contains .I and has maximal possible order, and then factors out the pullback of .J from the pullback of .I. It turns out that what is usually viewed as the main machinery, including the heavy one—maximal contacts, coefficient ideals, homogenization, independence of the embedding, etc.—generalizes quite easily to any setting, once an appropriate framework is chosen. And what we viewed as technical aspects in [7] and [5]—extending varieties to DM stacks, introducing an appropriate formalism of new centers, such as .(x 2 , y 3/2 ), etc.— seems to be the main choices which one has to carefully design. A wrong or insufficient choice often leads to an “almost” algorithm which gets stuck at an unexpected innocently, or technically looking, point. We finish Sect. 2 with an illustration of these general principles on the case of the classical algorithms—first we show how far one can go without boundaries, and then add this additional layer to the framework. In this case, one gets stuck because the order can jump on the maximal contact, so one has to consider order reductions of non-maximal order, and Hironaka’s insight was that this can be done once the excess of the exceptional divisor is well controlled by the boundary. If one uses precise weighted centers, the exceptional divisor is cleared off in a more precise way, and this explains why the basic weighted algorithm uses no boundary (or log structure) at all; this is the only method known so far which does not use log structures. A familiarity with the classical methods is assumed in our exposition, so we refer to chapter [16] and to the usual sources, such as [23] and [39]. Also, a very short and clear exposition (with some proofs omitted) can be found [12].

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1.2.2 The Logarithmic Methods In the next two sections we explain in detail the logarithmic analogue of the classical method, which was constructed in [7]. In Sect. 3 we proceed as much as one can within the framework of log varieties (as the reader can imagine, this was precisely our first line of research when working on [7]). The slogan of this part is to add “log” everywhere: log varieties, log smoothness, log smooth functoriality, log derivations and the associated notions of log order, log maximal contact and log coefficient ideal. The main novelty is that one can consider centers, which are powers of the centers of the form p p .(t1 , . . . , tn , u 1 , . . . , u r ), where .ti are regular parameters and .pi are arbitrary monomials. It turns out that in this fashion one almost obtains a perfect algorithm, which fails only at one point—this time the failure also happens in the log order reduction of nonmaximal log order d, but, in addition, the problem only pops up when the log order of .I is infinite. The solution this time is to allow blowings up of Kummer centers defined also by Kummer monomials like .up/d . This requires to extend both the formalism of such ideals and of their blowings up. The first task is solved in the Kummer étale topology and the second one forces one to consider stacks and non-representable modifications. The theory of Kummer centers and blowings up is developed in the beginning of Sect. 4, and then the same logarithmic algorithm constructed earlier works perfectly well. Section 5 is devoted to extending the absolute logarithmic methods to morphisms, following [8]. In fact, precisely the same algorithm works once one replaces absolute log derivations by relative ones. The only serious novelty is that one should take base changes into account. On the positive side, the algorithm is compatible with arbitrary base changes with a log regular source—a new type of functoriality. However, there is no free lunch, and another new feature is that the algorithm can fail, and in order for it to succeed one has to modify the base first. Non-surprisingly, once again the failure can happen at the “simple” monomial stage. Much more surprisingly is that we could not find a simple way to prove a monomialization theorem, which guarantees that the monomial step succeeds after a large enough base change, see Sect. 5.3. The existing proof is non-canonical, and we expect further progress to be possible. 1.2.3 The Weighted Methods Finally, in Sect. 6 we construct weighted algorithms which blow up arbitrary .Q-regular (or weighted) centers both in the non-logarithmic and logarithmic settings. We follow [5] in the non-logarithmic case and we follow [30] in the logarithmic ones. Note also that the same algorithm was constructed by McQuillan in [25], but the presentation uses a much more coordinate dependant language, so it is rather far from ours. The main idea is that q q we would like to blow up centers like .γ = (t1 1 , . . . , tn n ) which might lead to a singular output, but this can be resolved by blowing up an appropriate root .γ 1/n , obtaining a smooth stack-theoretic refinement of .Blγ (X). In order to make sense of things like .γ 1/n one has to introduce a new formalism of generalized ideals. In fact there exist a few ways to deal with this—valuative .Q-ideals, .Q-ideals (which are equivalent to Hironaka’s characteristic

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exponents) and Rees algebras, and we discuss them and relations between them in the first three subsections of Sect. 6. The paradigm of blowing up general .Q-regular ideals leads to what we call dream algorithms which introduce a simple invariant—the weighted (log) order, and improve it by a single blowing up along an .I-admissible .Q-regular center of maximal possible weighted (log) order. This results in what we call dream algorithms which require no history and simply repeat the same basic operation of weighted (log) order reduction. In addition, one obtains a non-embedded resolution which acts in the absolutely same manner—one simply blows up the unique maximal .Q-regular center contained in the scheme, and this blowing up improves the invariant. The logarithmic weighted algorithm is constructed very similarly but using the logarithmic setting. The main difference is that one also has to add a monomial part to the .Q-regular center, and one should take such a part as small as possible.

1.2.4 Conventiones Unless stated otherwise, we will always work over a ground field k of characteristic zero. By a blowing up we always mean a morphism .X = BlI (X) → X with the ideal .I ⊆ OX being part of the datum. Thus, the same morphism can underly different blowings up. By a slight abuse of language, saying that a morphism f is a blowing up without specifying the center we always mean that f underlies a blowing up along some center (in particular, it is projective). All log schemes are fs, see [37].

2

General Principles

In this section we will discuss principles and features shared by all known functorial resolution algorithms. In particular, we choose a presentation which might look a bit strange to the reader familiar only with the classical algorithm, but it extends naturally to other settings. We will end the section with a short description and re-interpretation of the classical algorithm in the new framework.

2.1

Frameworks

2.1.1 Modifications By a modification we mean a proper morphism .X → X which establishes an isomorphism  .U = U of dense open subschemes (subspaces, substacks, etc.) This definition applies to non-reduced objects as well, though we will usually work with generically reduced ones. For example, a blowing up is a modification if and only if its center is nowhere dense.

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2.1.2 Basic Choices of an Algorithm Each algorithm makes a few basic choices that we list below and call the framework of the algorithm. (0) The category .C of geometric objects the algorithm deals with, that will be called spaces, and the corresponding topology. For example, varieties over a fixed or varying fields, schemes with enough derivations, analytic spaces, stacks, log schemes, etc. The topology can be Zariski, étale, analytic, etc. (1) The class of regular spaces that will be called manifolds, and the class of regular morphisms. For example, smooth varieties, regular schemes, log smooth log varieties, etc., and smooth morphisms of varieties over a fixed field, regular morphisms between varieties over varying fields, log smooth morphisms of log varieties, etc. (2) The class of admissible modifications .f : X → X with X and .X manifolds. It will always be a variant of a blowing up along an admissible center .J (or simply a center), so we will use the notation .X = BlJ (X). In particular, the pullback .J OX is always an honest invertible ideal which defines the exceptional divisor .Ef ⊂ X . The center itself is an ideal in an appropriate topology, which can be rather fancy. For example, Kummer étale topology or h-topology. (3) A primary invariant which takes values in a totally or partially ordered set and more or less classifies different types of admissible centers. We call it the order of the center and extend to arbitrary ideals as follows: the order .ordX (I) of .I on X is the maximal order of an .I-admissible center, that is, an admissible center .J such that .I ⊆ J . Examples include the order of ideal, the log order and the weighted order .(d1 , . . . , dn ) of a weighted center .(t1d1 , . . . , tndn ). (4) A theory of derivations on manifolds. This amounts to choosing large enough sheaves of derivations one works with. For example, k-derivations or absolute derivations over .Q. Remark 2.1 (i) Choices (0)–(2) will be called the basic framework of the method. Choices (3) and (4) seem to be dictated by the basic framework, at least to a large extent. (ii) Concerning the choice of admissible blowings up, the general principle is that one should try to choose as large a class as possible with the restriction that the centers .J should possess a simple explicit description. In the cases we know, it seems that the framework essentially dictates a unique natural algorithm corresponding to it, and the larger the class of admissible blowings up is, the better algorithm one obtains. Even in the classical setting it is beneficial to consider centers of the form .J d , where .J = IV defines a smooth subvariety V . Since .BlJ d (X) = BlJ (X) this might look as a simple bookkeeping of the order d inside the center, but we will argue that in this form the description of the algorithm becomes both more natural and more similar to the logarithmic and weighted algorithms, e.g. see Remark 2.8.

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2.1.3 Functoriality All methods we will consider are functorial in the following strong sense: they are compatible with surjective regular morphisms. In particular, this implies that the method can be constructed étale locally (or even smooth locally), and once this is done the method globalizes by étale descent. This fact simplifies arguments tremendously as they become essentially of a local nature (see also Sect. 2.2.3). Compatibility with non-surjective regular morphisms holds on the level of a morphism, but not a finer structure of the principalization sequence of blowings up, and we will touch on this delicate issue later. Remark 2.2 (i) Historically, the first canonical algorithms (i.e. compatible with automorphisms) were constructed by Bierstone-Milman in [11] and Villamayor in [38], then Schwartz showed that Hironaka’s method is smooth functorial, see [31], and Włodarczyk was the first to emphasize on smooth functoriality and use it in an essential way in constructing the algorithm, see [39]. (ii) Smooth functoriality implies that the algorithm is equivariant with respect to any group scheme action, as any group scheme in characteristic zero is smooth.

2.2

Principalization and Resolution

The main result of each desingularization method is an appropriate principalization theorem, and as a consequence one obtains a non-embedded desingularization theorem.

2.2.1 Functorial Resolution Let .P be a class of regular morphisms (usually all regular morphisms in .C). By a .Pfunctorial resolution on .C we mean a rule .R which associates to each object X of .C a modification .R(X) : Xres → X with a regular source in such a way that .Yres = Y ×X Xres for any .P-morphism .Y → X in .C. The main resolution theorem for a given framework asserts that such a resolution exists. In addition, the desingularization morphisms .R(X) are projective (or non-representable global quotients of projective morphisms). In fact, they are naturally equipped with a structure of a composition of explicit blowings up, but this factorization is only compatible with surjective regular morphisms. 2.2.2 Functorial Principalization Let X be a manifold and .I an ideal on X. An admissible blowing up .X = BlJ (X) → X is called .I-admissible if .I ⊆ J . In such a case, .IOX is contained in the invertible ideal −1  .IE = J OX  , hence the transform .I = IOX  I E is defined. An .I-admissible sequence .f• : Xn X0 = X is a sequence of .Ii -admissible blowings up .fi : Xi+1 → Xi such that .I0 = I and .Ii+1 is the transform of .Ii . Such a sequence is called a principalization of .I if .In = OXn is trivial.

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By a .P-functorial principalization on .C we mean a rule which associates to any ideal I on a manifold X in .C a principalization .P(I) : Xn X0 = X of .I in such a way that for any .P-morphism .Y → X in .C and .I  = IOY the sequence .P(I  ) is obtained from the pullback .P(I)×X Y of .P(I) by omitting all trivial blowings up. The main principalization theorem asserts that such a principalization exists.

.

Remark 2.3 If .Y → X is surjective, then .P(I  ) = P(I)×X Y , but in general the blowings up along centers whose image in X is disjoint from the image of Y are pulled back to trivial blowings up and hence ignored.

2.2.3 Synchronization The above remark indicates that the principalization algorithm is not of local nature in the strict sense. For example, if .X = ∪ni=1 Xi is an open covering, one cannot reconstruct .P(I) from .P(I|X1 ), . . . , P(I|Xn ) without an additional synchronization data—what are  the trivial blowings up we removed after the restriction. However, if .X = ni=1 Xi and   .I = IOX  , then all these blowings up are kept in the sequence .P(I ), and as we remarked earlier .P(I) is easily reconstructed from .P(I  ). Informally speaking, when principalizing  .I the method has to compare the singularities of .I|Xi and decide which one is blown up earlier (or simultaneously), entering trivial blowings up at the other places. This is precisely the needed synchronization datum. Remark 2.4 (i) The above argument shows that it is important to consider simultaneous principalization on disconnected manifolds, and the method is only “local up to disjoint unions” or quasi-local accordingly to the terminology of [9]. (ii) Another way to establish a synchronization of local constructions is by use of an explicit invariant, for example, as [11] do. In fact, using the trick with disjoint unions is equivalent to the use of an abstract invariant, see [35, Remark 2.3.4].

2.2.4 The Re-embedding Principle There is one more important functoriality property satisfied by all known methods called the re-embedding principle. We say that a principalization method .P on .C satisfies the re-embedding principle if for any closed immersion of manifolds of constant codimension    .X → X an ideal .I on .X and its preimage .I ⊆ OX the blowing up sequence .f• = P(I) is obtained by pushing forward the blowing up sequence .f• = P(I  ), that is, the centers   .Ji of .f• are the preimages of the centers .J of .f• and (by induction on the length) each i   .X → Xi is the strict transform of .X → X. i

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2.2.5 Reduction to Principalization In all settings the appropriate desingularization theorem is a relatively easy corollary of the principalization theorem. Loosely speaking the general principle can be formulated as follows: Principle 2.5 If there exists a .P-functorial principalization on .C which satisfies the reembedding principle, then there exists a .P-functorial desingularization .R on the class of locally equidimensional generically reduced spaces from .C which locally possess a closed immersion into a manifold. The embeddability assumption is automatic for varieties, formal varieties or analytic spaces, and is only relevant for general excellent schemes. The local equidimensionality condition is used to construct an embedding of constant codimension. The argument in all settings is essentially the same: to resolve a space Z, one locally embeds it into a manifold X and constructs the resolution of Z from the principalization of .IZ . Loosely speaking, before blowing up a generic point .η ∈ Z the principalization has to guarantee that it is a generic point of an admissible center, and in all methods this amounts to resolving the Zariski closure of .η. Moreover, because of the codimension assumption, all generic points of Z are blown up simultaneously at some blowing up .Xi+1 → Xi and its center contains a component which is the strict transform of Z. In particular, .Zi → Z is the induced desingularization of Z. Independence of the embedding in all methods is proved by use of the re-embedding principle and a simple computation showing that an embedding of minimal possible codimension is unique étale locally (or formally locally). Remark 2.6 (i) The desingularization morphism .Zi → Z is naturally a composition of blowings up .Zj +1 → Zj , .0 ≤ j ≤ i − 1 with centers .Jj OZj . In the classical resolution each center .Vj = V (Jj ) is smooth, but the intersection .Vj ×Xj Zj can be singular. In particular, the factorization of .Zi → Z into a composition of blowings up is not too informative. (ii) So-called strong resolution methods construct a resolution which is a composition .Zi Z0 = Z of blowings up along smooth centers. Perhaps the main advantage of this is that for any closed immersion .Z → Y into a manifold the resolution automatically extends to a modification of manifolds .Yi → Y with .Zi a closed subscheme in .Yi : just consider the pushout .Yi Y of the sequence .Zi Z. (iii) The only known method to construct strong resolution is to force the condition that .Vj ⊆ Zj in the principalization sequence, and hence the whole sequence .Xi X is the pushforward of the sequence .Zi Z. In the classical case this is achieved by serious additional work building on the usual principalization (the so-called presentation of the Hilbert-Samuel function in [11]). We will see that in the weighted desingularization methods strong factorization is achieved just as a by-product.

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2.2.6 The Miracle The reduction of resolution to a seemingly very different principalization problem is usually viewed as a brilliant trick if not a miracle. Nevertheless, we claim that this is not so surprising. An equivalent formulation of existence of resolution is that manifolds are cofinal among the set of all modifications of a generically reduced space X. When one studies modifications of a manifold X, it is hard (if not impossible) to explicitly describe all modifications .X → X with .X a manifold, so it is natural to only consider basic explicit modifications of this form—admissible blowings up and their compositions. The principalization theorem asserts that for any ideal .I there is an admissible sequence  .X X which principalizes .I, and by the universal property of blowings up this happens if and only if the morphism .X → X factors through .BlI (X). By Chow’s lemma blowings up form a cofinal family of modifications of X, hence the principalization just asserts that admissible sequences form a cofinal family of modifications of a manifold. In this form, it is rather natural to expect that the theorems are related and the principalization theorem is finer. Remark 2.7 One may also wonder if the following weak factorization conjecture holds: any modification of manifolds can be factored into a composition of admissible blowings up and blowings down. This conjecture provides the next level of depth. In the classical case the only known argument deduces it with a large amount of work from .Gm equivariant principalization in the next dimension. In other settings this is still open, though we expect that an analogous approach with birational cobordisms and .Gm -equivariant principalization should work there too. It would be especially interesting to check this for semistable models and morphisms.

2.2.7 Order Reduction In first approximation, the principalization is achieved by successive order reduction procedures: reduce the order .d = ordX (I) of .I by blowing up centers of order d. In the non-weighted algorithms one reduces this problem to an order reduction on a maximal contact hypersurface. However, the order can jump under this reduction, so for inductive reasons one also has to solve the problem of reducing the order of .I below e only by blowings up e-centers for any fixed value .e ≤ d of the invariant. This results in the accumulation of exceptional divisors in the transform, and one has to use a log structure to control this—guarantee that the exceptional divisor is monomial and deal with it mainly by combinatorial methods. In fact, this is the only place in the algorithms, where some flexibility can take place. In weighted algorithms the order reduction is done by a single weighted blowing up, so they are what we call dream algorithms—no history is needed, each blowing up is independent of the rest and reduces the invariant further. However, the argument that a unique maximal .I-admissible center exists is rather complicated and, again, uses the theory of maximal contact and homogenization. In particular, it completely fails in positive characteristic.

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The Classical Algorithm: A First Attempt

The classical method was already discussed in chapter [16], so we assume that the reader is familiar with the main ideas and constructions, and our goal is to briefly re-interpret it within the general paradigm we described earlier in this section. Later we will develop the logarithmic algorithm pretty much in the same spirit. In Sect. 2.3 we will discuss what can be done without the boundary and where this attempt fails. However, all constructions we are going to describe are relevant, and in the next subsection, we will just add the boundary as an additional layer of the framework. For simplicity, we work with k-varieties.

2.3.1 The Framework One considers the category .C of varieties over k, manifolds are smooth varieties and the algorithms will be smooth functorial. Admissible centers are of the form .J = IVd , where .IV ⊂ OX is the ideal of a submanifold .V ⊂ X and .d ≥ 1. We call such a center a d-center. An admissible blowing up is the usual blowing up of the center. The derivation theory is (≤d) given by the sheaves .DX = DerX/k of k-derivations and the sheaves .DX of differential k-operators of order at most d. The primary invariant of the center is just the multiplicity d of .I d .1 The order of an arbitrary ideal .I at x is the maximal d such that .Ix ⊆ IVd . Clearly, it suffices to take the center .V = {x}, and then we obtain the usual definition of the order. Remark 2.8 Classically one only considers 1-centers, works with marked ideals .(I, d) and uses a d-transform after blowings up along a smooth center V that lies inside the locus of points where the order is at least d. This is equivalent to our admissibility condition d .I ⊆ J = I V and using the usual transform with respect to the blowing up along .J . So, we just provide a slightly different interpretation of the classical constructions.

2.3.2 Derivations Derivations of ideals provide a convenient way to describe all basic ingredients of the algorithm (except the boundary): (≤d)

(1) The maximal order .ordX (I) of .I on X is the minimal number d such that .DX (I) = OX . The order of .I at a point .x ∈ X is computed similarly. (2) A maximal contact to .I at x is any closed smooth subscheme .H → X which locally (≤d−1) (Ix ). at x is of the form .V (t) with .t ∈ DX

1 We use the notion of the multiplicity of a center instead of the order to avoid confusion with the general order of ideals it is used to define. This is justified because the classical multiplicity of .V (I d )

at any its point is d.

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(3) The homogenized coefficient ideal is the homogenized weighted sum of derivations, which are weighted by their orders: 

C(I) =

d−1 

.

a∈Nd

:

d−1 i=0

(≤i)

DX (I)

ai

.

ai (d−i)≤d! i=0

Remark 2.9 Usual coefficient ideals are defined using only the corresponding powers of the derivations of .I, but the homogenized version is integral over it, and hence can be used instead. The homogenized coefficient ideals were introduced by Kollár, see [23, §3.54]. They subsume the homogenization procedure of Włodarczyk, see [39, §2.9].

2.3.3 Order Reduction If .e ≤ d = ord(I), then an order e-reduction of .I is an .I-admissible sequence of blowings up along e-centers .Xn X0 = X such that .ordXn (In ) < e. Remark 2.10 Usually one talks about order reduction of a marked ideal .(I, d) by blowing up smooth centers, and d indicates which power of the exceptional divisor to factor out on each transform. The two languages are equivalent.

2.3.4 The Maximal Order Case The main loop of classical principalization iteratively performs order reduction with .e = d—the so-called maximal order case. In this case, the theory of maximal contact implies that for any maximal contact H (which exists locally) pushing out from H to X establishes a one-to-one correspondence between order d-reductions of .I and order .d!-reductions of .C(I)|H , so we can apply induction on dimension. Moreover, for any other maximal contact .H  the restrictions of .C(I) to H and .H  can be taken one to another by an étale correspondence, hence the construction is independent of choices and globalizes. 2.3.5 General Order e-Reduction It can happen that .ordH (C(I)|H ) > d! and so the induction forces one to also consider the non-maximal order case. The trick is to reduce this to the maximal order case by controlling the accumulated exceptional divisor, and for this job one has to add one more layer to the framework—the boundary.

2.4

The Classical Algorithm: The Boundary

2.4.1 The Framework In fact, instead of manifolds X one works with pairs .(X, E), where the boundary (or the exceptional divisor) E is an snc divisor. In some versions, one also orders components of E by a history function. One restricts the set of the admissible centers .J = IVd by requiring

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that V has simple normal crossings with E, and then the boundary .E  on .X = BlJ (X) is combined from the preimage of E (the old boundary) and the exceptional divisor of   .X → X (the new boundary). This guarantees that .E is snc. However, one still has to struggle with two complications mainly caused by the fact that one uses all derivations rather than those that preserve E, so all constructions are not well-adapted to E and one has to fix this essentially by hand. Fortunately, this can be done by two tricks.

2.4.2 Removing the Old Boundary The first complication is that a maximal contact H does not have to be transversal to E, so .E|H does not have to be a boundary. This is resolved by separating H and the old boundary by iterative order reduction of .I along the maximal multiplicity strata of the old boundary. This trick forces one to introduce a secondary invariant s—the number of the old boundary components remaining since a maximal contact was created. As a result, the (non-normalized, see below) total invariant is .(d0 , s0 ; d1 , s1 , . . . ) rather than just the string of orders .(d0 , d1 , . . . ). 2.4.3 The Normalized Degrees  In addition, one usually normalizes the orders by .qi = di / j · · · > dn−1 ≥ d > dn .

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The Initial Cleaning Step This step makes the log order finite and consists of the single blowing up along the d-center .J (d) , where .J = M(I)1/d . Clearly, .J (d) is the integral closure of .M(I), hence the blowing up has the same effect and the transform .I1 is of finite log order by Lemma 3.25. In particular, .N1 is trivial. The Regular Step This step accepts .Ii = Iicln Ni with .di ≥ d as an input and outputs a dsequence .f•d : Xi+1 = Ym Y0 = Xi , which is associated in the sense of Lemma 3.31 to the .di -sequence .f•di : Ym Y0 = Xi which reduces the log order of .Iicln . By Lemma 3.31 cln N the sequence outputs .Ii+1 = Ii+1 i+1 with .di+1 < di . d The .di -sequence .f• i is constructed as follows. Working étale locally we can assume that there exists a maximal contact H to .I and by induction assumption the ideal (d!) .C(I)|H possesses a log order .di !-reduction .g• : Hm H0 = H with centers .J j . We define .f•di to be the sequence with centers .Jj(d) , where .Jj is the preimage of .J j under the surjection .OYj  OHj . Thus, if locally on .Yj we have that .Hj = V (t) and the center of .Hj +1 → Hj is .(t1 , . . . , tr , up1 , . . . , ups )(dj !) , then the center of p p (d ) d .Yj +1 → Yj is .Jj = (t, t1 , . . . , tr , u 1 , . . . , u s ) j and the corresponding center of .f• is p p (d) s 1 .(t, t1 , . . . , tr , u , . . . , u ) . By Theorem 3.28 the sequence .f•d is indeed a log order d-reduction of .Ii , and by Theorem 3.29 it is independent of choices and descends from the étale local construction. The Final Cleaning Step This is trivial—we just repeat the cleaning step, but this time 1/d M(In ) = Nn , so we just blow up the d-center .Nn = (Nn )(d) . Thus, .X = Xn and cln .In+1 = In is as required. .

4.2.4 Justification of the Algorithm It is clear from the construction that the obtained sequence .X → X is a log order dreduction for .I—it first makes the log order finite, then reduces the log order of the clean part below d, and then removes the invertible monomial part. Compatibility of this algorithm with log smooth morphisms .Y → X follows from the fact that all basic ingredients of our construction are log smooth functorial, as was stated in Corollaries 3.24 and 3.20, Lemma 3.14(ii), etc. Finally, it suffices to check the re-embedding principle étale-locally, so we can assume that .X → Y is a closed immersion of log orbifolds of pure codimension one and .I  is the preimage of .I in .OY . Then .logordY (I  ) = 1 and X is a maximal contact to .I  , so by the regular step of the algorithm .F(I  , 1) is the pushforward of .F(I, 1), which is precisely what is claimed by the re-embedding principle. Remark 4.14 We already noted that the log principalization is obtained by applying the log order reduction with .d = 1. Similarly to the proof of log order reduction, one can principalize just a bit faster by iteratively applying maximal order reductions. This might look more natural as there is no need to do the final cleaning step, but the drawback is that such an algorithm does not satisfy the re-embedding principle.

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4.2.5 The Invariant We constructed the log order reduction .F(I, d) as a composition of blowing up sequences cln .Xi+1 Xi , which can be parameterized by the strictly decreasing sequence .logord(I i ). In particular, the first stage is numbered by .∞, the last stage—by a number which is strictly smaller than d and will be replaced by 0 for convenience, and all regular steps are numbered by a finite number .di ≥ d. The first and last steps are single monomial blowings up, and each regular step corresponds to the log order .di !-reduction of .C(Iicln )|Hi . We define the invariant of a separate blowing up of .F(I, d) by induction on .dim(X) as follows: the blowing up of the first step (if non-trivial) has invariant .(∞), the blowing up of the last step (if non-trivial) has invariant .(0), the blowings up of the i-th regular sequence have invariant .(di , invHi ), where .invHi denotes the invariant of the corresponding log order cln .di !-reduction sequence of .C(I i )|Hi . Each non-normalized invariant is of the form .(d0 , d1 , . . . , dn , ∗), where .d0 ≥ d, .di+1 ≥ di ! and .∗ ∈ {∞, 0}, and by induction on the length of the invariant and the fact that cln .logordX (I i ) strictly decreases, we obtain that each separate blowing up in the sequence i indeed decreases the value of the invariant. Hence the same is also true for the normalized  invariant .(q0 , q1 , . . . , qn , ∗), where .qi = di / j 0 then we will use n q q qi the suggestive notation .(f1 1 , . . . , fn n ) = i=1 (fi ) to denote the valuative ideal .γ = min1≤i≤n qi vfi . (ii) Assume now that X is smooth. If .γ is such that locally on X there exists a presentation q1 qn .γ = (t , . . . , tn ) such that the support .V (t1 , . . . , tn ) of .γ is smooth of codimension 1 n, then .γ is called a .Q-regular center on X. Such a center is called a smooth weighted center if one can choose a presentation with .qi = 1/wi for each i, and a smooth weighted center is reduced if in addition .(w1 , . . . , wn ) = 1. The tuple .w = (w1 , . . . , wn ) is called the tuple of weights. If the support of a .Q-regular center is of codimension 2 and higher, then there are many different ways to choose the regular parameters. For example, .V (x, y q ) = V (x + y n , y q ) for any natural .n ≥ q. However the weights are well defined. Exercise 6.7 Show that the multiplicities .q1 , . . . , qn ∈ Q>0 are uniquely determined by q q a .Q-regular center .γ = (t1 1 , . . . , tn n ).

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6.1.5 Blowings Up of Valuative Q-Ideals To any valuative .Q-ideal .γ ∈ Q,+ (X) on a normal variety X one associates the graded .OX -algebra .Rγ = ⊕d≥0 Rd called the Rees algebra of .γ and defined by .Rd (U ) = {f ∈ OX (U )| vf ≥ dγ |U }, where .U = RZ(U ) ⊆ X. Then the blowing up of X along .γ is the stack-theoretic Proj of the Rees algebra: .Blγ (X) = Proj X (Rγ ). Remark 6.8 Each .Rd is an ideal on X, which is the pushforward of the d-th power of the ideal .(γ OX )d . So our definition of the Rees algebra and its blowing up is a precise analogue of the definition of Kummer blowings up in Sect. 4.1.8. Exercise 6.9 Let X be a normal variety with a valuative .Q-ideal .γ of the special form 1/d 1/d γ = (f1 1 , . . . , fn n ) (this is a .Q-ideal or an idealistic exponent from Sect. 6.2 below).

.

(i) The algebra .Rγ is integrally closed and finitely generated over .OX . (ii) If .γ = vI for a usual ideal .I ⊆ OX , then .Rγ = ⊕d∈N I (d) = (⊕d∈N I d )nor is the integral closure of the usual Rees algebra of .I. In particular, .Blγ (X) = BlI (X)nor and it is singular even for the valuative center associated with .I = (t1 , t22 ). (iii) .γ corresponds to a usual invertible ideal on .X = Blγ (X).

6.1.6 Smooth Weighted Blowings Up By a smooth weighted blowing up of X we mean blowing up of a smooth weighted center. Such blowings up output smooth stacks, unlike the blowings up along an arbitrary .Qreduced center. The following result is established by a direct chart computation (e.g. see [4, §3.6]). Exercise 6.10 1/w

1/w

(i) Let .w = (w1 , . . . , wn ) ∈ Nn≥1 be a tuple of weights and .γ = (t1 1 , . . . , tn n ) a smooth weighted center on X. Then .X = Blγ (X) is a smooth DM stack whose stabilizers on the i-th chart are subgroups of .μwi . The smooth weighted center .γ becomes a usual invertible ideal on .X which will be denoted .γ OX . 1/dw 1/dw (ii) If .d ≥ 1 and .γ  = d −1 γ = (t1 1 , . . . , tn n ), then .Bld −1 γ is the root stack obtained from .Blγ (X) by extracting the d-th root from .γ OX .

6.1.7 Associated Weighted Blowings Up The weighted algorithms use certain .Q-regular centers including all those with natural multiplicities, which correspond to usual ideals. Blowing up such a center can output a singular variety, hence we should use a stack-theoretic refinement instead. The trick is to blow up an appropriate root of the center. q

q

Definition 6.11 Assume that .γ = (t1 1 , . . . , tn n ) is a .Q-regular center. Chose the representation .qi = ai /bi with .(ai , bi ) = 1 and let .a = lcm(a1 , . . . , an ). Then .a −1 γ =

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(t1 1 , . . . , tn n ) is the smooth weighted center associated with .γ and .X = Bla −1 γ (X) is the associated weighted blowing up. We will use the notation .X = wBlγ (X). q /a

q /a

Remark 6.12 The blowing up .Blγ (X) is a partial coarsening of the weighted blowing up .wBlγ (X). The scaling factor .a = lcm(a1 , . . . , an ) is chosen so that the stack-theoretic refinement .wBlγ (X) is smooth and the stacky structure is increased as little as possible.

6.2

Q-Ideals and Idealistic Exponents

In principle, the formalism of valuative .Q-ideals introduced in [5] covers our needs. This section will not be used in the sequel, but it is quite enlightening to also study the smaller class of .Q-ideals which really play a role in this story, especially because they formalize the classical notions of idealistic exponents and marked ideals. These notions are well behaved only on normal schemes, so we restrict to this generality. We also refer to [30, §2.2] and [40, §2.1].

6.2.1 Q-Ideals A .Q-ideal on a normal scheme X is a valuative .Q-ideal which locally on X is of the form 1/d1 1/d −1 .γ = min(d , . . . , fn n ). In i vfi ) where .fi are functions. In other notation, .γ = (f1 particular, a .Q-regular center is a .Q-ideal. In a sense, .Q-ideals generalize usual ideals in the same meaning as valuative .Q-ideals generalize valuative ideals, and the following exercise formalizes this point of view. Exercise 6.13 (i) Let .γ be a valuative .Q-ideal. Show that it is a usual ideal (i.e. .γ = vI for an ideal .I) if and only if .γ is both a valuative ideal and a .Q-ideal. In particular, .γ is a .Q-ideal if and only if .dγ is an ideal for some d. (ii) Show that the multiplicative monoid of .Q-ideals is uniquely divisible and hence it is the divisible hull of the monoid of integrally closed ideals. In particular, .Q-ideals can be safely presented in the form .I 1/d , where .I is an ideal (or its integral closure). (iii) A marked ideal .(I, d) can be viewed as the .Q-ideal .I 1/d : show that this correspondence agrees with the usual operations on marked ideals—inclusion, multiplication and summation (homogenized or not). (iv) Give an example of a valuative ideal, which is not a .Q-ideal. (Hint: one can take .X = Spec(k[x, y]) and .γ = max(0, min(vx , vy − vx ))—the valuative ideal corresponding to the ideal .(x, y/x) on .Bl(x,y) (X). Then the minimal .Q-ideal containing .γ is .min(vx , vy ).)

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Remark 6.14 (i) In fact, the notion of a .Q-ideal is nothing else but a formalization of Hironaka’s notion of idealistic exponent. As we saw, this can be done in two ways—either realize them as valuative .Q-ideals of a special form, or as roots of usual ideals considered up to integral closure. (ii) The classical order reduction of a marked ideal .(I, d) can now be formalized as reducing the order of the .Q-ideal .I 1/d below 1 by blowing up smooth centers .J and factoring out their pullback. Our interpretation in Sect. 2.3 is that one reduces below d the order of .I itself by blowing up centers .J d . These interpretations just differ by normalization.

6.2.2 Normalized Blowings Up Normalizing blowings up of .Q-ideals to stay in the category of normal schemes one obtains the following universal property, see [29, Theorem 3.4.3]. Theorem 6.15 If .I 1/d is a .Q-ideal on a normal variety X and .X = BlI 1/d (X)nor is the normalized blowing up, then .I 1/d OX is an invertible ideal and .f : X → X is the universal morphism of normal stacks with this property: if .g : T → X is a morphism of normal stacks such that .g −1 (X \ V (I)) is dense in T , then there exists at most one factorization of g through f and it exists if and only if .I 1/d OT is an invertible ideal. In particular, we obtain a generalization of Exercise 6.10(ii). Remark 6.16 This universal property immediately implies that the normalized blowing up along .I 1/d can be described using the usual normalized blowing up .Y = BlI (X)nor → X and the normalized root stack construction .Bl(IOY )1/d (Y ) → Y : the first makes the pullback of .I invertible and the second extracts the d-th root.

6.3

Rees Algebras and Rees Blowings Up

In order to describe non-embedded resolution in the most precise way, we should also consider non-normal varieties and the weighted blowings up induced on them from weighted blowings up of the ambient manifold. This theory was developed by Rydh in his work on Nagata compactification for stacks and later by Quek-Rydh in [29]. One has to switch to the language of arbitrary (non-normal) Rees algebras. The reader can safely skip (or just look through) this section; it will only be used to formulate the non-embedded weighted resolution in the most precise way.

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6.3.1 Rees Algebras The .Q-ideals can also be interpreted in terms of another classical object—Rees algebra. As in [29], by a Rees algebra on a variety X we mean a finitely generated graded .OX -algebra R = ⊕d∈N Id t d ⊆ OX [t]

.

such that .I0 = OX and the ideals .Id form a decreasing sequence: .In ⊆ Im for .m ≤ n (the latter condition is automatic when R is normal). By the support of R we mean .V (R) = V (I1 ). For any .n ≥ 1 we have that .I1n ⊆ In ⊆ I1 , hence .V (In ) = V (R). If R is normal, then necessarily X is normal and .I1 is integrally closed. Conversely, if X is normal then the normalization .R nor of a Rees algebra is also a Rees algebra on X (for general schemes one should also assume that X is quasi-excellent, or, at least, Nagata). In particular, for any valuative .Q-ideal .γ on a normal scheme X we have that .Rγ is a normal Rees algebra. The natural construction in the opposite direction associates to a Rees algebra R the valuative .Q-ideal .γR = mind≥1 d −1 vId . In fact, it is a .Q-ideal which possesses a concrete description in term of generators: if R is generated over .OX by .f1 t d1 , . . . , fn t dn , then .γR = min(d1−1 vf1 , . . . , dn−1 vfn ). Exercise 6.17 Show that for any Rees algebra R on a normal variety X one has that .γR is the minimal valuative .Q-ideal whose Rees algebra equals R. Deduce that the constructions .γR and .Rγ provide a one-to-one correspondence between normal Rees algebras and .Qideals on X. Also deduce that for any valuative .Q-ideal .γ with .R = Rγ one has that .γR is the maximal .Q-ideal such that .γR ≤ γ . Recall that the blowing up of a valuative .Q-ideal .γ is determined by its Rees algebra R = Rγ , hence it coincides with the blowing up along the .Q-ideal .γR . Thus, the theory of weighted blowings up of normal varieties can be described entirely in terms of .Q-ideals or Rees algebras. To extend this to non-normal varieties one should use non integrally closed Rees algebras.

.

6.3.2 Pullbacks For any morphism of normal schemes .Y → X and a .Q-ideal .I 1/d on X one defines its pullback .(I OY )1/d which we will also denote as .I 1/d OY for shortness. It is easy to see that this operation is well defined. Also, for any morphism of schemes .Y → X and a Rees algebra .R = ⊕d Id on X one defines the pullback .ROY = ⊕d Id OY . 6.3.3 Rees Blowings Up To any Rees algebra R on a variety X one associates the blowing up .: X = BlR (X) = ProjX (R). It comes equipped with the exceptional divisor .E  whose invertible ideal will be denoted .IE  . Namely, for .ad t d ∈ Rd the restriction of .IE  onto the corresponding chart d −1 ])/G ] corresponds to the .G -equivariant ideal .(t −1 ), where .t −1 = .[Spec(R[(ad t ) m m d−1 d /ad t ∈ R[(ad t d )−1 ]. ad t

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6.3.4 The Universal Property of Weighted Blowings Up Weighted blowings up possess a certain universal property similar to the one satisfied by the usual blowings up but also more subtle. The proof is a direct computation, see [29, Theorem 3.2.9]. Theorem 6.18 Let X be a variety and R a Rees algebra on X with the Rees blowing up  .f : X = BlR (X) → X and support .V = V (R). Then: (0) f is an isomorphism over .X \ V . (i) .Id OX ⊆ IEd  with equality holding for any divisible enough d (in fact, it suffices that d is divisible by the weights of a set of generators). (ii) If .h : T → X is a morphism such that .h−1 (X \ V ) is schematically dense, then the category of factorizations of h through .X is equivalent to the set of invertible ideals d .IT such that .Id OT ⊆ I with equality holding for any divisible enough d. T

6.3.5 Strict Transforms If .Y → X is a morphism of varieties and R is a Rees algebra on X with .V = V (R), then the strict transform of Y with respect to the blowing up .X = BlR (X) → X is the schematic closure of the preimage of .X \ V in .Y ×X X . As in the classical case, the universal property easily implies the following description of strict transforms, see [29, Corollary 3.2.14]. Corollary 6.19 Let .Y  → X be the strict transform of a morphism .h : Y → X with respect to a Rees blowing up .X = BlR (X) → X. Then .Y  = BlR OY (Y ). In addition,      .Y = Y ×X X whenever h is flat, and .h : Y → X is a closed immersion whenever h is a closed immersion.

6.3.6 Q-Regular Centers Finally, we would like to define .Q-regular centers in an arbitrary variety Z. For a tuple .f = (f1 , . . . , fn ) of functions .fi ∈ (OZ ) and a tuple of positive rational numbers .q = (q1 , . . . , qn ) consider the Rees algebra .Rf,q = ⊕d Id with each .Id generated by all monomials .f1l1 . . . fnln with .l1 /q1 + . . . + ln /qn ≥ d. By a .Q-regular center of multiplicity .q = (q1 , . . . , qn ) on Z we mean a Rees algebra which is locally of the form .Rf,q so that for any .z ∈ V (R) the images of .f1 , . . . , fn in .mz /m2z are linearly independent and the image of the set {f1l1 . . . fnln | l1 /q1 + . . . + ln /qn < 1}

.

in .Oz /I1 Oz is linearly independent. By the associated weighted blowing up we mean the blowing up of the Rees algebra obtained by shifting the weights to .a −1 q where .a ≥ 1 is the minimal natural number

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such that each .qi /a is of the form .1/wi . Locally this algebra is generated by the elements f1 t w1 , . . . , fn t wn . These definitions are compatible with closed immersions into manifolds:

.

Exercise 6.20 Keep the above notation and fix a closed immersion .Z → X with a smooth X. (i) Show that R extends to a .Q-regular center on X in the following sense: choose any family of parameters .t1 , . . . , tn on X which restricts to .f1 , . . . , fn and consider the .Qq q regular center .γ = (t1 1 , . . . , tn n ). Then .R = Rγ OZ . In particular, the Rees blowing up of Z along R is the strict transform of .Blγ (X) → X. (ii) Moreover, show that geometrically speaking .V (γ ) is contained in Z in the sense that .γ ≤ vI for .I = IZ . (iii) Conversely, show that any .Q-regular center .γ on X such that .γ ≤ vI restricts to a .Q-regular center on Z. q q (iv) Finally, show that if Z is normal, then .R nor corresponds to the .Q-ideal .(f1 1 , . . . , fn n ).

6.4

Non-logarithmic Weighted Algorithms

In this section we will outline the simplest dream principalization—the one without boundary.

6.4.1 Weighted Framework The geometric objects are just DM stacks of finite type over k, and one uses the usual smoothness and sheaves of derivations .DX = Derk (OX , OX ). Admissible centers are .Qregular centers, and such a center .γ is .I-admissible if .γ ≤ vI (which informally means that .I ⊆ γ ). Admissible blowings up are weighted blowings up associated with such centers. 6.4.2 Weighted Order To complete the framework it remains to classify the centers and introduce an invariant. For this we always order parameters so that the tuple of multiplicities is monotonically increasing: .q1 ≤ q2 ≤ · · · ≤ qn . Naturally, the invariant .w-ord(γ ) of the center is defined to be the ordered tuple .q = (q1 , . . . , qn ), and we provide the set of invariants with the lexicographic order, where shorter sequences are larger, for example, .(2) > (2, 2) (alternatively, one can finish each string with .∞, making the invariant more similar to the classical one). Exercise 6.21 Check that the invariant is monotonic: if .γ ≤ γ  , then .w-ord(γ ) ≤ w-ord(γ  ).

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For an arbitrary proper ideal .I we define the weighted order w-ordX (I) = max w-ord(γ ),

.

γ ≤vI

where the maximum is over all .I-admissible .Q-regular centers. The weighted order at a point is defined by taking the minimum over all neighborhoods: w-ordx (I) = min w-ordU (I|U ).

.

x∈U

For completeness we also define the invariant in extreme cases: .w-ord(OX ) = (0) and w-ord(I) = () if .V (I) contains a generic point of X.

.

6.4.3 Weighted Order Reduction Now we can formulate the main properties of the weighted framework, which completely dictate what the algorithm is. In fact, it turns out that there is a unique candidate for the first weighted blowing up and already this blowing up reduces the weighted order. Theorem 6.22 Let X be a smooth DM stack of finite type over k, and let .I be an ideal on X. (i) The weighted order .w-ordX (I) = (q1 , . . . , qn ) satisfies the following integrality  condition: .q1 ∈ N and .qi+1 ij =1 (qj − 1)! ∈ N for .2 ≤ i ≤ n. (ii) There exists a unique .I-admissible center .γ such that .w-ord(γ ) = w-ordX (I) and .V (γ ) contains all points .x ∈ X with .w-ordx (I) = w-ordX (I). (iii) Let .γ be the center described by (ii), .X = wBlγ (X) the associated weighted blowing up (Definition 6.11) and .I  = (γ OX )−1 IOX the transform of .I. Then  .w-ordX  (I ) < w-ordX (I). (iv) The center depends smooth functorially on the pair .(X, I).

6.4.4 Weighted Principalization Part (i) of the theorem implies that the set of invariants of .w-ordX is a well-ordered subset of .{0} ∪ (∪n∈N Qn>0 ), therefore iterating the weighted order reduction we arrive at the end to the only ideal with zero invariant—the unit ideal. This yields the main result about weighted principalization. Theorem 6.23 There exists a principalization method .F which associates to any ideal .I on a smooth DM stack X of finite type over k a sequence of smooth weighted blowings up  .F(I) : X = Xn X0 = X such that the following conditions are satisfied. (i) The sequence is a principalization: .In = OXn and each .Xi+1 → Xi is .Ii -admissible, where .I0 = I and .Ii+1 is the transform of .Ii for .0 ≤ i ≤ n − 1.

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(ii) The sequence depends smooth functorially on the pair .(X, I): for any smooth morphism .Y → X with .J = IOY the sequence .F(J ) is obtained from the pullback sequence .F(I) ×X Y by omitting all empty weighted blowings up. (iii) The method requires no history (I call this a dream method): each weighted blowing up .Xi+1 → Xi depends only on .(Xi , Ii ), but not on .(Xj , Ij ) with .j < i.

6.4.5 Examples When the Dream Fails In the classical setting no dream principalization algorithm exists. Usually one argues that blowing up a pinch point on the Whitney umbrella .Z = V (x 2 − y 2 z) → A3 = X creates another pinch point, hence without memory one goes into an endless loop. However, blowing up X along .V (x, y) resolves the singularity, which makes the case not fully convincing. Here is Włodarczyk’s favorite example which avoids this. Let .X = A4 = Spec(k[x, y, z, t]) and .Z = V (x 2 − yzt), in particular, .G = S3 acts on it by permuting .y, z, t. The singular locus of Z is the union of the y, z and t-axes, and the origin O is the only .S3 -invariant smooth center which contains O and lies in the singular locus. As in the case of Whitney umbrella, blowing O up creates another singularity of the same type (in fact, 3 singularities on different charts permuted by .S3 ). This shows that there is no memoryless smooth functorial principalization which only blows up smooth centers. Exercise 6.24 Show that the weighted order at 0 is .(2, 3, 3, 3), the dream algorithm blows up .(x 1/3 , y 1/2 , z1/2 , t 1/2 ) and the weighted order drops after this blowing up.

6.4.6 Weighted Desingularization As in the classical case, if .Z → X is a closed generically reduced substack of constant codimension, then the principalization of .IZ ⊂ OX blows up all components of the strict transform of Z at the same blowing up .Xn+1 → Xn . This implies that the strict transform .Zn → Z of Z is smooth, hence .Zn → Z is a resolution. By the usual argument, uniqueness of minimal embeddings up to an étale correspondence yields a smooth functorial desingularization method for locally-equidimensional and generically reduced stacks of finite type over k. 6.4.7 Back to Schemes The stacky structure can be removed by a destackification. As a result one obtains a new desingularization method, which is more efficient than the classical one, but does not possess new theoretical properties. However, we will now see that a minor modification leads to a so-called strong desingularization method. 6.4.8 Strong Weighted Desingularization When .Z → X is resolved by the classical principalization of .IZ the i-th center .Vi lies in the preimage of Z, but it does not have to be contained in the strict transforms .Zi → Xi of Z, and the center .Vi |Zi = Vi ×Xi Zi of .Zi+1 → Zi is usually singular, see [12, Example 8.2]. The same is true for the weighted principalization: it can happen

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that .V (γi )  Zi . However, if .X = wBlγ (X) → X is the weighted order reduction of     .I = IZ , .Z → X is the strict transform of Z, and .I is the transform of .I, then .Z is a   closed subset of .V (I ), and hence .w-ordX (IZ  ) ≤ w-ordX (I ) < w-ordX (IZ ). Therefore, the more economical way to resolve Z is to proceed at the second stage with .IZ  instead of .I  , etc. In such a way we do not achieve principalization of .IZ but just successively modify the strict transform of Z. Moreover, not only the i-th center .γi is contained in the i-th strict transform .Zi (see Exercise 6.20), it depends only on the pair .(Xi , Zi ) and then the re-embedding principle implies that in fact it only depends on .Zi . This provides the following non-embedded dream algorithm. Theorem 6.25 There exists a desingularization method .R which associates to any locally equidimensional generically reduced DM stack Z of finite type over k a sequence of weighted blowings up with .Q-regular centers .R(Z) : Z  = Zn Z0 = Z such that the following conditions are satisfied. (i) The sequence is a desingularization: .Z  is smooth. (ii) The sequence depends smooth functorially on Z: for any smooth morphism .Y → Z the sequence .R(Y ) is obtained from the pullback sequence .R(Z) ×Z Y by omitting all trivial blowings up. (iii) The method requires no history: each weighted blowing up .Zi+1 → Zi depends only on .Zi .

6.4.9 The Geometric Interpretation The algorithm iteratively repeats the same base step: chose a canonical .Q-regular center .RZ depending only on Z and blow it up. This .RZ is in fact the unique .Q-regular center whose multiplicity .(q1 , . . . , qn ) is maximal possible and which contains all points of Z with this invariant. Furthermore, this center witnesses the non-smoothness of Z by the fact that it cannot be extended to a thicker center with larger multiplicities. Blowing up this obstacle we kill it, and it turns out that the blown up variety has a smaller maximal non-smoothness obstacle. q

q

Example 6.26 Take .Z = Spec(k[t1 , . . . , tn ]/(t1 1 + . . . + tn n )) with .2 ≤ q1 ≤ q2 ≤ · · · ≤ q q qn . The maximal center in this case is induced by the usual ideal .I = (t1 1 , . . . , tn n ) and l1 its normalized Rees algebra .R = ⊕d Id , where .Id is generated by monomials .t1 . . . tnln  with . j lj /qj ≥ d and .I1 = I nor . Clearly, .V (I1 ) → Z and the associated weighted blowing up is a stack theoretic refinement of the blowing up along .I1 . Let also .I>1 be  the ideal generated by the monomials with . j lj /qj > 1. Then the non-smoothness of Z is detected by the fact that the closed immersion .V (I1 ) → Z does not extend to a q q / I>1 . square-zero thickening .V (I>1 ) → Z because .t1 1 + . . . + tn n ∈

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6.4.10 Relation to the Theory of Maximal Contact Now let us briefly discuss the justification. As we saw, everything follows easily from Theorem 6.22. At first glance one might expect that such a result, once correct, should be provable in a few ways, including rather direct ones. The following example by Włodarczyk shows that one should be more careful because such a .Q-regular center does not exist in positive characteristic, hence an argument should be subtle enough and use the zero characteristic assumption. Example 6.27 The example is again. . . a Whitney umbrella .I = (x 2 + y 2 z) on .X = Spec(k[x, y, z]), but this time when k is perfect of characteristic 2. As in characteristic zero, the weighted order at the origin is .(2, 3, 3), but this time the whole singular locus .C = V (x, y) consists of pinch points because there is an automorphism .(x, y, z) → (x + ty, y, z + t 2 ) which translates C. Clearly, the local centers exist in this case at any point of C, but they do not glue to a global one. Moreover, at the generic point .η ∈ C the invariant is .(2, 2) and the trouble hides in the fact that the unit .z ∈ Oη is not a square. Note ¯ viewed as a point of .Xk(z) also that the invariant at the geometric generic point .η, ¯ , is again .(2, 3, 3). All in all, we see that no nice theory of weighted centers seems to be possible, and, maybe, imperfect residue fields and units whose value at a point is not a p-th power should somehow be taken into account. My personal expectation is that the formalism of characteristic exponents might be useful but insufficient. Given our current knowledge, when working on [5] and [30] the fastest way was to use the classical maximal contact theory both to construct such a center and prove its uniqueness. We do expect that a more direct argument should exist, and looking for it is one of the future projects.

6.4.11 Construction of the Center By the theory of maximal contact, working locally at a point .x ∈ X we define an iterated sequence of maximal contacts to iterated coefficient ideals. Formally speaking, we obtain a neighborhood .H0 of x, a partial sequence of regular parameters .t1 , . . . , tn and a sequence of ideals .Ii on .Hi = V (t1 , . . . , ti ) of orders .di = ordx (Ii ) such that .I0 = I|H0 , each .ti |Hi is a maximal contact to .Ii , .di = ordHi (Ii ) for any .0 ≤ i ≤ n, in the notation of Sect. 3.3.4 we have that Ii+1 =

d i −1

.

≤a (DX (Ii ), di − a) |Hi+1

a=0

is the restriction of the coefficient ideal of .Ii onto .Hi+1 for any .0 ≤ i ≤ n−1, and .In = 0. In particular, .H0 is small enough so that each order accepts its maximum at x.

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Once the above choices are done, setting .qi = di−1 / .q1 = d0 ) one obtains that

i−2

j =0 (dj

− 1)! (in particular,

w-ordH0 (I) = w-ordx (I) = (q1 , . . . , qn )

.

q

q

and the center on .H0 is .γ = (t1 1 , . . . , tn n ). Note that this datum is very close to the one produced by the classical algorithm: the first blowing up of the non-weighted principalization is along the center .(t1 , . . . , tn ) and .(q1 , 0; . . . ; qn , 0; ∞) is the invariant of Bierstone-Milman for this blowing up. Certainly, the above sketch only provides a construction and explains the integrality condition and smooth functoriality, but one has to prove that, indeed, .γ is .I-admissible, of maximal weighted order, and unique. All this is done using standard results from the theory of maximal contact (e.g. [5, Lemma 4.4.1]) and homogenization (needed to pass from one maximal contact to another). We refer the reader to the proof of [5, Theorem 5.1.1] for details. These ideas are also recalled in chapter [4], and in chapter [40] the weighted principalization is constructed with all proofs and details in the context of cobordant blowings up, when instead of stack-theoretic blowings up one considers the canonical presentation by a torus quotient. Essentially this means that the orbifolds we consider are replaced by their torus equivariant representable covers, hence, being smooth local, the arguments and proofs are essentially the same.

6.5

Logarithmic Weighted Algorithms

The simplest dream algorithm does not use any log structures. In particular, one obtains a principalization of an ideal, which is just an invertible ideal, with an arbitrarily singular support. In addition, one does not obtain resolution of divisors by snc ones. There is a logarithmic refinement of the dream algorithm which addresses these issues. It should certainly exist in the relative setting, but so far the theory was only developed by Quek in [30] in the absolute case—the case of log varieties. This method carefully combines the logarithmic and weighted settings and many things have to be checked, but the main line is to imitate the dream algorithm when working with log varieties and using arbitrary weighted submonomial centers. The reader already saw these ideas, so we will only explain the main novelty.

6.5.1 The Framework Naturally, one works with logarithmic DM stacks of finite type over a field, logarithmic derivations and logarithmic smoothness. Admissible centers on a log orbifold X are valq q uative .Q-ideals which are locally at .x ∈ X of the form .γ = (t1 1 , . . . , tn n , up1 , . . . , upr ), where .t1 , . . . , tm is a regular family of parameters at x, and .p1 , . . . , pr ∈ (M x )Q are rational monomials. Admissible blowings up are weighted blowings up of such centers

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q

as we define below. We will use the shorter notation .γ = (t1 1 , . . . , tn n , Nγ ), where p p .Nγ = (u 1 , . . . , u r ) is a Kummer monomial ideal called the monomial type of .γ . As in the non-logarithmic case, one has a certain freedom in choosing the regular parameters .t1 , . . . , tn and the other data is fixed: Exercise 6.28 Show that both the multiplicity .(q1 , . . . , qn ) of .γ and the monomial type are uniquely determined by .γ .

6.5.2 Weighted Submonomial Blowings Up As in the non-logarithmic case, by a weighted submonomial center we mean an admissible center with .qi = 1/wi for natural weights .wi ∈ N. For an admissible center .γ = q q (t1 1 , . . . , tn n , Nγ ) with .qi = ai /bi and .a = lcm(a1 , . . . , an ) the center .a −1 γ is weighted and we define the weighted blowing up along .γ to be .X = wBlγ (X) = Bla −1 γ (X). For any ideal .I on X such that .γ is .I-admissible the transform .I  = (γ OX )−1 IOX is defined. Again, a direct chart computation shows that a weighted submonomial blowing up of a log orbifold X outputs a log orbifold .X , see [30, Lemmas 4.1 and 4.2]. 6.5.3 Weighted Log Order and Monomial Type q q For an admissible center .γ = (t1 1 , . . . , tn n , Nγ ) set .wlogordX (γ ) = (q1 , . . . , qn ) if .Nγ = ∅ (i.e. there is no monomial part) and .wlogordX (γ ) = (q1 , . . . , qn , ∞) if .Nγ is non-empty. This invariant and the monomial type provide the following monotonicity: if .γ < γ  , then  .wlogordX (γ ) ≤ wlogordX (γ ) and in the case of equality one has that .Nγ  Nγ  . For an arbitrary ideal .I on X we define the weighted log order by wlogordX (I) = max wlogord(γ ).

.

γ ≤vI

In addition, .wlogord(OX ) = (0) and .wlogord = () if .V (I) contains a generic point. Furthermore, we define the monomial type .NI of .I to be empty if .wlogord(OX ) does not end with .∞ and to be the minimal (with respect to inclusion) monomial type of an .I-admissible .γ such that .wlogordX (I) = wlogord(γ ).

6.5.4 Weighted Log Order Reduction Now we can formulate the main theorem about weighted submonomial centers. The only new tool when comparing it to the non-logarithmic analogue is that one should also pay attention to the monomial type of an ideal. Theorem 6.29 Let X be a log smooth DM stack of finite type over k, and let .I be an ideal on X.

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(i) The weighted log order .wlogordX (I) = (q1 , . . . , qn , ∗) and the monomial type .NI are well defined and the following integrality condition is satisfied: .q1 ∈ N and i .qi+1 j =1 (qj − 1)! ∈ N for .2 ≤ i ≤ n, and .∗ is either .∅ or .∞. (ii) There exists a unique .I-admissible center .γ such that for each point .x ∈ X with .wlogordx (I) = wlogordX (I) one has that .wlogordx (γ ) = wlogordx (I) and .Nx (γ ) = Nx (I). (iii) Let .γ be the center described by (ii), .X = wBlγ (X) the associated weighted blowing up and .I  = (γ OX )−1 IOX the transform of .I. Then .wlogordX (I  ) < wlogordX (I). (iv) The center depends log smooth functorially on the pair .(X, I).

6.5.5 Weighted Log Principalization As in the non-logarithmic case iterating the weighted log order reductions one obtains a dream algorithm for log principalization. Theorem 6.30 There exists a log principalization method .F which associates to any ideal .I on a log smooth DM stack X of finite type over k a sequence of weighted blowings up  .F(I) : X = Xn X0 = X such that the following conditions are satisfied. (i) The sequence is a log principalization: .In = OXn and each .Xi+1 → Xi is .Ii admissible, where .I0 = I and .Ii+1 is the transform of .Ii for .0 ≤ i ≤ n − 1. (ii) The sequence depends log smooth functorially on the pair .(X, I): for any log smooth morphism .Y → X with .J = IOY the sequence .F(J ) is obtained from the pullback sequence .F(I) ×X Y by omitting all trivial weighted blowings up. (iii) The method requires no history: each weighted blowing up .Xi+1 → Xi depends only on .(Xi , Ii ).

6.5.6 Strong Logarithmic Resolution The same argument as in the non-logarithmic case shows that applying weighted log order reduction to the strict transforms of .Z → X one obtains the following non-embedded dream log desingularization algorithm Theorem 6.31 There exists a desingularization method .R which associates to any locally equidimensional and generically log smooth log DM stack Z of finite type over k a sequence of weighted blowings up .R(Z) : Z  = Zn Z0 = Z such that the following conditions are satisfied. (i) The sequence is a log desingularization: .Z  is log smooth. (ii) The sequence depends log smooth functorially on Z: for any smooth morphism .Y → Z the sequence .R(Y ) is obtained from the pullback sequence .R(Z) ×Z Y by omitting all trivial blowings up.

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(iii) The method requires no history: each weighted blowing up .Zi+1 → Zi depends only on .Zi .

6.5.7 Justification As in the non-logarithmic case described in Sect. 6.4.11, one uses the logarithmic theory of maximal contact (and the sheaf of logarithmic derivations .DX ). Locally at a point .x ∈ X one defines an iterated sequence of logarithmic maximal contacts to the iterated coefficient ideals, obtaining a neighborhood .H0 of x, a partial sequence of regular parameters .t1 , . . . , tn and a sequence of ideals .Ii on .Hi = V (t1 , . . . , ti ) of orders .di = ordx (Ii ) such that .I0 = I|H , each .ti |Hi is a logarithmic maximal contact to .Ii , .di = ordHi (Ii ) for any .0 ≤ i ≤ n − 1, we have that Ii+1 =

d i −1

.

≤a (DX (Ii ), di − a)|Hi+1

a=0

is the restriction of the coefficient ideal of .Ii onto .Hi+1 for any .0 ≤ i ≤ n − 1, and logord(In ) = ∞. Then the monomial type .NI0 is defined as the monomial ideal generated by the monomials from the monomial hull .M(In ) of .In . Once these choices are done, the invariant and the center are read off as follows:

.

wlogordH0 (I) = w-ordx (I) = (q1 , . . . , qn , ∗),

.

 where .qi = di−1 / i−2 j =0 (dj − 1)! (in particular, .q1 = d0 ), .∗ = ∅ if .NI0 = ∅ and q1 qn .∗ = ∞ otherwise, and the center on .H0 is .γ = (t , . . . , tn , NI ). The justification 0 1 is analogous, but it works with the logarithmic theory of maximal contact and also addresses the monomial type in the end. Once again we can relate this datum to the first blowing up of the non-weighted logarithmic principalization: the latter blows up the center 1/q1 .(t1 , . . . , tn , N I0 ) and the invariant at this step is .(q1 , . . . , qn , ∗), see Remark 4.15.

7

Resolution for Quasi-Excellent Schemes and Other Categories

Throughout the notes we only worked with schemes and stacks of finite type over k. In this section we will discuss resolution in the wider context of quasi-excellent schemes and other categories, such as formal schemes, and complex or non-archimedean analytic spaces. We only aim to provide a very brief survey of some tools and literature. In addition, we will discuss directions which were not fully verified so far, but we expect the results to be true. The two main methods we will discuss are via extending the framework, and via black box reduction to (appropriate) quasi-excellent schemes. Note that a rather detailed survey on the second approach can be found in [34] and we will refer to it from time to

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time. Unfortunately [34] reflects our knowledge from more than 10 years ago, and hence considers only the questions of extending the classical methods to wider settings.

7.1

Reduction to Quasi-Excellent Schemes

In this subsection we will show that desingularization of objects that look completely non-algebraizable nevertheless follows from desingularization of quasi-excellent schemes. Thus, quite surprisingly, resolution of singularities seems to be a purely algebraic phenomenon. The key observation is that the latter desingularization should be functorial with respect to all regular morphisms. Also we will see in Sect. 7.3 that within the class of qe schemes there exist various bootstraps which allow to essentially reduce the problem to resolving algebraic varieties (these methods were used in [32] and [35]). We start with specifying the classes of geometric objects whose desingularization we will discuss, and then we will discuss the reduction to qe schemes.

7.1.1 Analytic Spaces Resolution of singularities makes sense in various categories of geometric objects, where appropriate notions of smoothness and blowings up make sense. Probably, the most natural case is that of complex analytic spaces, and Hironaka himself outlined how his proof should be modified in that case. In [11] Bierstone-Milman tried to axiomatize various situations in which their desingularization algorithm applies. In particular, they claimed that it applies to algebraic spaces of finite type over a field, analytic spaces—complex, real and non-archimedean, and certain quasi-analytic objects. This is definitely true with our current knowledge, though I cannot track the original argument in some cases (one cannot use Zariski topology when working with algebraic spaces; also in Berkovich geometry sheaves of differentials are defined with respect to the G-topology rather than the usual one). 7.1.2 Quasi-Excellent Schemes Next, let us discuss the case of schemes not of finite type over a field. Hironaka used formal completion in his original work [19], so he had to prove his results for the much wider is class of schemes of finite type over local rings A such that the homomorphism .A → A regular. Recall that a morphism .Y → X of noetherian schemes is regular if it is flat and has geometrically regular fibers. For morphisms of finite type this notion is equivalent to smoothness, so regularity is the natural generalization of smoothness in the case of large

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morphisms. Soon after this Grothendieck defined a more general class of quasi-excellent or qe rings2 by two conditions: X is qe if the following two conditions are satisfied: (N) After Nagata: for any integral Y of finite type over X the singular locus of Y is open. (G) After Grothendieck (though we saw, that historically it should have been named after X,x is regular. Hironaka): for any .x ∈ X the completion homomorphism .OX,x → O Some discussion and examples related to these properties and their relation to resolution can be found in [34, §2.3]. The main observation of Grothendieck was [18, .IV2 , Proposition 7.9.5] claiming that even a weakest consistent resolution theory is possible only for qe schemes: if any integral X-scheme of finite type possesses a desingularization, then X is qe. Already Grothendieck expressed a hope that any integral qe scheme possesses a resolution and this is widely believed to be true. In characteristic zero, this was proved in [32], and stronger versions are in [35] and [36].3 Thus, when working with schemes we will always restrict to the quasiexcellent schemes of characteristic zero. By Grothendieck [18, .IV2 , Proposition 7.8.6(i)] if X is excellent, then any X-scheme of finite type is excellent too, and the same argument applies to qe schemes as well.

7.1.3 Stacks A stack is called qe if it has a smooth cover by a qe scheme. We tacitly used in these notes that any smooth-functorial algorithm automatically extends from schemes of finite type over a field to stacks of finite type over a field. The same argument applies to qe schemes and stacks. Also, any method which involves stack-theoretic blowings up (e.g. logarithmic or weighted principalization) should be developed in the generality of qe stacks right ahead. 7.1.4 Formal Schemes We say that a noetherian formal scheme .X = (X, OX ) is a formal variety if its closed fiber .Xs = (X, OX /I ), where I is the maximal ideal of definition, is an algebraic variety. Naturally, when working with more general formal schemes we will have to impose a quasi-excellence restriction. A formal scheme X is qe if its closed fiber is a qe scheme. This definition is really useful because of the following theorem of Gabber, see [24]: an I -adic notherian ring A is qe if and only if .A/I is qe. This is a deep fact whose proof uses a weak local resolution of singularities. In particular, it implies that quasi-excellence is preserved by formal completions, and hence also by formal localizations, passing from

2 The terminology was introduced later. Grothendieck only used the notion of excellent schemes to

denote universally catenary quasi-excellent ones. 3 In general only resolution of qe threefolds is known, see [14].

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A to .At, etc. Part of the difficulty stems from the fact that the G-property along is not satisfied even by passing from A to .At. Gabber’s theorem implies that if X is qe and Y is of topologically finite type over Y , then Y is qe. Before the theorem was available one had to work with clumsier and ad hoc definitions, e.g. see [32, §3.1]. An important property of qe formal schemes is that formal localization homomorphisms .A → A{f } are regular. The latter condition is extremely important, since it allows to extend to qe formal schemes notions from the theory of schemes which are local with respect to the topology of regular covers. In particular, one can define the notions of singular loci, see [32, §3.1], and regular morphisms. In particular, .Y → X is regular if it is covered by morphisms .Spf(Bij ) → Spf(Ai ) with regular homomorphisms .Ai → Bij . The general principle is very simple—use the usual definition which works with rings in the affine case and use compatibility of the notion with regular morphisms to globalize. Blowings up of formal schemes are defined in a similar fashion, see [32, §2.1]. If .X = Spf(A) with an I -adic A, then one simply blows up .Spec(A) and I -adically completes, and in general one glues the local construction using that blowings up are compatible with flat morphisms. In particular, this construction is well defined for arbitrary noetherian formal schemes.

7.1.5 Geometric Spaces For concreteness we will work with one of the following spaces: (1) (2) (3) (4)

Quasi-excellent schemes. Quasi-excellent formal schemes. Complex analytic spaces. Non-archimedean analytic spaces introduced by Berkovich, see [10]. Certainly there are other possibilities—rigid or adic analytic spaces, Nash spaces, etc.

7.1.6 Affinoid Spaces In Sect. 7.1.4 we saw how constructions from the scheme theory can be transferred to formal schemes. It turns out that one can study analytic spaces similarly. This is not surprising in the non-archimedean setting, since Berkovich spaces are pasted from affinoid spaces .M(A), which are spectra of k-affinoid algebras. Affinoid algebras are excellent and for any subdomain embedding .M(B) → M(A) the homomorphism .A → B is regular by results of Ducros, see [15] or [3, §§6.2–6.3]. Nevertheless, an analogous theory also exists for complex analytic spaces. Each such space X is covered by so-called semi-algebraic Stein compacts .Xi (for example, closed subdomains of polydiscs), the ring .Ai = OX (Xi ) of overconvergent functions on .Xi is excellent and controls .Xi well enough. Again, if .Xj ⊆ Xi , then the homomorphism .Aj → Ai is regular. In detail, these claims are checked in [3, Appendix B], and a relative GAGA

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over Stein compacts is established in [3, Appendix C]. In particular, GAGA implies that, as in the formal case, complex blowings up are obtained by analytifying the algebraic ones.

7.1.7 Reduction to qe Schemes The general principle is very simple: any desingularization, principalization, etc. method .F, which is functorial with respect to arbitrary regular morphisms, automatically extends to analytic spaces and qe formal schemes. The construction is as follows: (0) Cover the space X with affinoid/affine subspaces .Xi corresponding to qe rings .Ai . If X comes equipped with an ideal .I ⊆ OX one also obtains an ideal .Ii ⊆ Ai , etc. Also, cover each .Xi ∩ Xj with affinoid/affine subspaces .Xij k . (1) Consider the blowing up tower .F(Spec(Ai )) of .Spec(Ai ) and apply the analytification/completion functor to produce a blowing up tower of .Xi . This step uses the relative GAGA as we need to analytify ideals on blowings up in the tower. (2) Show that the obtained towers .F(Xi ) are compatible on intersections because .F respects the regular morphisms .Spec(Xij k ) → Spec(Xi ). In each case some minor details should be spelled out. In particular, this reduction scheme was used in [35, 36], and [3, §6], but it is the latter reference where all details were carefully spelled out (in the case of the factorization functor), a relative GAGA for complex spaces was constructed, etc.

7.1.8 Non-compact Objects Finally, we note that functorial methods can be also extended to non-compact spaces, including locally noetherian qe schemes. Clearly, the only way to do this is to cover X by affine subspaces .Xi and glue the blowing up sequences for different .Xi . The technical complication is that the resulting sequence can be infinite; for example, this happens  when .X = ∞ i=1 Xi and the resolution of .Xi takes i steps. However, one can naturally define infinite ordered sequences of blowing up (called hypersequences in [35, §5.3]) with the following local finiteness condition: over any compact subspace of X the sequence contains only finitely many non-trivial blowings up. For such a sequence a composition is well defined, and gluing local resolutions of the subspaces .Xi one obtains a hypersequence of blowings up of X in general, see [35, §5.3].

7.2

Extending the Framework

The second method to extend algorithms developed for algebraic varieties is to directly adopt them to another category. For example, this is what Bierstone and Milman did for the classical method and the categories of analytic spaces. This line of research is very natural, as it just explores in which generality the methods work. Nevertheless, it seems that it is almost not presented in the literature, so I will just express my expectations. In brief,

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absolutely in line with the philosophy of frameworks I expect that once the framework of the method extends to a wider setting, the method extends as well. So, one should construct appropriate blowings up, generalized ideals and derivation theory. In case of nonembedded resolution one should also worry that embedding into a smooth space exists. Note also that the theories of orbifolds and logarithmic analytic spaces are folklore and were partially developed in the literature. Now, let us discuss case by case.

7.2.1 Analytic Spaces I expect that, excluding resolution of morphisms, all methods we discussed in these notes extend to complex and non-archimedean analytic spaces. One should use the usual analytic differentials and derivations. Perhaps the shortest way to introduce weighted blowings up is via .Q-ideals and Rees algebras. I do not expect any serious complication. As for principalization on relative orbifolds and resolution of morphisms of analytic spaces, there is one major obstacle—the monomialization theorem, see Sect. 5.3. For example, let .B = P2C , .B0 = A2C and .Z = V (x − ey ) is a curve in .B0 which cannot be extended to B. Let .X = A1B0 = A3C with coordinates .x, y, z and .I the ideal on X given by .(x − ey , z). Then principalization of .I over .B0 goes by blowing up V in .B0 , which increases the log structure by .up = x − ey and then blowing up the ideal .(x, up ). However, once trying to principalize .I on .X → B one is stuck. I expect that except the monomialization the whole algorithm works fine. As for monomialization there are two hopes/questions which, at the very least, do not contradict any example we know: Does the monomialization theorem holds true when .X → B is proper? Does the monomialization theorem holds for an arbitrary smooth morphism  .X → B of smooth analytic spaces if one allows base changes .B → B of the following  more general form: .B → B is a cover for the topology generated by modifications and open covers? 7.2.2 Schemes with Enough Derivations Arbitrary qe schemes X may have too nasty absolute derivation theory. Not only, the sheaf .DX = DerQ (OX , OX ) does not have to be quasi-coherent, already for .X = Spec(R) with R a DVR it can happen that the sheaf .DX has a zero stalk at the closed point, see [34, Example 2.3.5(ii)]. Certainly, all methods of these notes cannot apply to schemes with such derivation theory, so one should consider only qe schemes whose derivation theory is reach enough. The following definition was suggested already in [35, Remark 1.3.1(iii)]: a scheme X has enough derivations if for any point .x ∈ X all elements of the cotangent space 2 .mx /mx are distinguished by elements of .DX,x . In other words, the homomorphism 2 ∗ .DX,x → (mx /mx ) to the tangent space is onto. Note that since .DX is not quasi-coherent, .DX,x can be strictly smaller than .Der(OX,x , OX,x ) so this condition asserts that there exist enough derivations in a neighborhood of x. It can be shown that schemes with enough derivations are qe and their class is closed under passage to schemes of finite type. I expect that non-logarithmic principalization methods—classical and weighted, work for general

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schemes with enough derivations (in the classical case this was conjectured in [35]) and are functorial with respect to arbitrary regular morphisms. Similarly, I expect that the nonembedded methods work for schemes which are étale-locally embeddable into regular schemes with enough derivations. I expect that the same holds true for logarithmic methods and log schemes with enough log derivations, where the latter means that log derivations distinguish both regular and monoidal parameters. In the non-weighted case this was proved in [8] as the particular case of [8, Theorem 1.2.6] when the target B is just .Spec(Q). In fact, [8] is the only paper I am aware of, where the notion of enough derivations was seriously explored. It even studied resolution of morphisms .X → B with enough relative derivations, and showed that it works whenever .dim(B) ≤ 1. In the case of a higher dimensional B a much stronger restriction on derivations is needed—the so-called abundance of derivations, which also takes into account derivations in “constant directions” which distinguish elements of a transcendence basis of .k(x)/k(b). This condition cannot be borrowed to analytic spaces, and this explains why our monomialization theorem does not extend to analytic spaces when .dim(B) > 1. Finally, we note that, beyond algebraic varieties, the most important case of schemes with enough derivations are schemes of finite type over noetherian complete local rings. We will later see, why this class (or its certain subclasses) are so important.

7.2.3 Formal Schemes The treatment of formal schemes should be analogous to schemes, namely, I expect that the methods extend to formal schemes with enough (log) derivations. The most important case is that of formal varieties.

7.3

Desingularization for Quasi-Excellent Schemes

Finally, let us describe the situation with arbitrary qe (formal) schemes. The tools we have described so far do not suffice to establish resolution for them, and it seems that the only way is to use a certain descent from the formal completion via the (G) property. Thus, at first stage one should construct a desingularization or principalization method .FC on the class .C of (formal) schemes of finite type over complete local rings or its suitable subclasses (for example, by the method of Sect. 7.2), and at the second step one applies descent. The classical solution uses descent of open ideals, and necessarily constructs a new method .F, which is more complicated even on the schemes from .C. This method is called localization or induction on codimension, it goes back to Hironaka’s original paper, and (due to my ignorance) it was re-invented in [32]. A natural alternative would be to try to descent more general ideals, since this would just extend .FC from .C to the class of all qe schemes. The only known result in this direction is McQuillan’s proof that weighted centers indeed satisfy this descent, and hence the weighted resolution and principalization algorithms extend to arbitrary qe schemes, see [25, Section VII].

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7.3.1 Induction on Codimension The method is very robust and applies to all algorithms, see [19, Chapter IV, §1] and [34] (see also [35, §4] and [36, §3.4]). In addition, this method is also used in the proof of the monomialization theorem. However, for the sake of concreteness we will only illustrate the idea on the case of principalization. Assume first that .X = Spec(A) is a local regular qe scheme and I is supported at the = Spec(A) at the maximal ideal m is closed point .s ∈ X. Then the formal completion .X The principalization .s ∈ X. a regular scheme and .I = I A is supported at the closed point 0 = X of .I , which exists by step one, only blows up centers contained in the n X .X i ⊆ OX .s. They correspond to ideals .J -adic topology, hence fiber over i open in the .m all these ideals algebraize and the tower is obtained by m-adic completion of a tower .Xn X0 = X of blowings up with centers .Ji ⊆ OXi . It then easily follows by descent that the latter sequence principalizes I . In the general case one proceeds by induction on codimension of the image of nonprincipalized locus in X. First, one considers the finite set of points .x1 , . . . , xn ∈ T0 =  V (I ) of minimal codimension, principalizes the pullback of I to . ni=1 Spec(OX,xi ) by the method from the previous paragraph and then just blows up the schematic closure of the centers, obtaining a blowing up sequence .X X which principalizes I over .x1 , . . . , xn . The centers do not have to be smooth over specializations of .x1 , . . . , xn , but by induction assumption we can assume that principalization (and hence also resolution) has been already constructed for them. Thus, one simply resolves each center before blowing it up. This inserts intermediate blowings up in the sequence .X X and the algorithm becomes more complicated, but this does not affect the situation over .x1 , . . . , xn . At the next step one considers the image .T1 ⊂ X of .V (I  ), where .I  ⊆ OX is the transform of I . It is a closed subset of .T0 \{x1 , . . . , xn }. Choose points .y1 , . . . , ym ∈ T1 of minimal codimension, find a sequence .X X which principalizes .I  over the preimages of .y1 , . . . , ym , etc. Remark 7.1 (i) As an input for the induction on codimension scheme it suffices to take a principalization method .FC which applies to regular schemes X of finite type over the spectrum .Spec(R) of a complete local ring and ideals .I ⊂ OX such that .V (I ) is contained in the preimage of the closed point. This is a serious restriction, which allows to obtain .FC via a black box construction from principalization of varieties. Indeed, such a scheme can locally be realized as a completion of a variety and I algebraizes to an ideal on the variety, so one can just pullback principalization on the algebraization. The subtle point in this method is independence of algebraization, which is based on a theorem of Elkik, see [34, §4]. This method was implemented in [35, §3], though nowadays I think that extending the framework to schemes with enough derivations would be a better solution since it is more robust and easy to generalize to other situations. (ii) In order to prove that the obtained method is functorial with respect to all regular morphisms one should prove that this is true for the input algorithm which operates

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with varieties. This turns out to be a bit subtle because there are regular morphisms between varieties defined over different fields k, and the classical method uses only k-derivations. One solution is to extend the framework by working with absolute derivations over .Q. However, there is also a black box argument which proves that the classical method (and all methods we constructed in these notes) are compatible with arbitrary regular morphisms between varieties. This was worked out in [13].

7.3.2 Direct Descent Finally, let us discuss a hypothetical direct descent. Assume, again that X is a local regular is its completion, but this time consider an arbitrary ideal I on X with qe scheme and .X comes from X makes it reasonable to hope that completion .I ⊆ OX . The fact that .I any regular functorial principalization of .I should descent to X. We illustrate this with a ×X X would be noetherian and qe, then the two pullbacks wrong argument: if .Y = X would coincide with the principalization of of the principalization to the completion .Y → X are regular in such case), and hence the principalization .I OY (the projections .Y would descend to X by the flat descent. Unfortunately, Y is usually very far from being noetherian so a completely different argument is needed, but the question if some other form of descent to X is possible seems very natural. I expect that if such descent works in the local case, then one should be able to patch the local solutions on an arbitrary qe X using the N-property.

Appendix: Integral Closure Recall that the integral closure .I nor of an ideal .I ⊆ A consists of all elements .t ∈ A  satisfying a monic equation .t n + ni=1 ai t n−i = 0 with .ai ∈ I i . Any ideal J with .I ⊆ J ⊆ I nor is called integral over I . Both notions are compatible with localizations and hence extend to sheaves of ideals on schemes. If .J is integral over an ideal .I on a scheme X and .Y → X is a morphism, then .J OY is integral over .IOY . Lemma A.1 Let X be a normal scheme with an invertible ideal .I. Then for any ideal .J with .I nor = J nor , one has that .I = J . In other words, .I is integrally closed and it is the integral closure only of itself. Proof The claim can be checked locally, so assume that A is a local normal domain, I = (t) is principal and J is an ideal with .I nor = J nor . If s is integral over I , then it  satisfies an equation .s n + ni=1 ai t i s n−i = 0 and hence .s/t satisfies a monic equation over A. By the normality of A we have that .s/t ∈ A, that is .s ∈ I . Thus, .I = I nor . / J , but it is integral over J and we take a monic Assume that .J = I nor = I . Then .t ∈  equation .t n + ni=1 bi t n−i = 0 with .bi ∈ J i ⊂ I i = (t i ) and minimal possible n. By the minimality of n each element .ci = bi /t i is not a unit, as otherwise .t i − bi /ci = 0 would

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be an equation of smaller degree. Hence we obtain that .t n u = 0, where .u = 1 + is a unit, and this yields a contradiction.

n

i=1 ci

 

The normality assumption is necessary in the lemma. For example, in .A = Spec(k[x 2 , x 3 ]) the principal ideal .(x 2 ) has a non-invertible integral closure .(x 2 , x 3 ). Corollary A.2 If .I and .J are ideals on a normal scheme X such that .J nor = (I d )nor , then .(BlI (X))nor = (BlJ (X))nor . Proof The claim easily follows from the two particular cases: when .J = I d , and when .J is integral over .I. In the first case, for any modification .Y → X we have that .J OY = (IOY )d is invertible if and only if .IOY is invertible. So, by the universal property of blowings up .BlI (X) = BlI d (X). If .J is integral over .I, then .J OY is integral over .IOY , so if Y is normal and one of these ideals is principle, then the other is (and both coincide)   by Lemma A.1. The second case follows.

References 1. D. Abramovich, Stacks for Everyone who Cares about Varieties and Singularities, in New Techniques in Resolution of Singularities (Springer, Cham, 2023), this volume 2. D. Abramovich, K. Karu, Weak semistable reduction in characteristic 0. Invent. Math. 139(2), 241–273 (2000). MR 1738451 (2001f:14021) 3. D. Abramovich, M. Temkin, Functorial factorization of birational maps for qe schemes in characteristic 0. Algebra Number Theory 13(2), 379–424 (2019) 4. D. Abramovich, M. Temkin, J. Włodarczyk, Birational geometry using weighted blowing up, in New Techniques in Resolution of Singularities (Springer, Cham, 2023), this volume 5. D. Abramovich, M. Temkin, J. Włodarczyk, Functorial embedded resolution via weighted blowings up, arXiv e-prints (2019), arXiv:1906.07106 6. D. Abramovich, M. Temkin, J. Włodarczyk, Toroidal orbifolds, destackification, and Kummer blowings up. Algebra Number Theory 14(8), 2001–2035 (2020). With an appendix by David Rydh. MR 4172700 7. D. Abramovich, M. Temkin, J. Włodarczyk, Principalization of ideals on toroidal orbifolds. J. Eur. Math. Soc. (JEMS) 22(12), 3805–3866 (2020). MR 4176781 8. D. Abramovich, M. Temkin, J. Włodarczyk, Relative desingularization and principalization of ideals, arXiv e-prints (2020), arXiv:2003.03659 9. K. Adiprasito, G. Liu, M. Temkin, Semistable reduction in characteristic 0, November (2018). http://arxiv.org/abs/1810.03131 10. V.G. Berkovich, Étale cohomology for non-Archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math. 78, 5–161 (1993). MR 1259429 (95c:14017) 11. E. Bierstone, P.D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math. 128(2), 207–302 (1997). MR 1440306 (98e:14010) 12. E. Bierstone, P.D. Milman, Functoriality in resolution of singularities. Publ. Res. Inst. Math. Sci. 44(2), 609–639 (2008). MR 2426359

Relative and Logarithmic Resolution of Singularities

217

13. E. Bierstone, P.D. Milman, M. Temkin, Q-universal desingularization. Asian J. Math. 15(2), 229–249 (2011). MR 2838221 14. V. Cossart, O. Piltant, Resolution of singularities of arithmetical threefolds. J. Algebra 529, 268– 535 (2019) 15. A. Ducros, Les espaces de Berkovich sont excellents. Ann. Inst. Fourier (Grenoble) 59(4), 1443– 1552 (2009). MR 2566967 (2011a:14048) 16. A. Fruhbis-Krüger, A computational view on Hironakas resolution of singularities, in New Techniques in Resolution of Singularities (Springer, Cham, 2023), this volume 17. J. Giraud, Sur la théorie du contact maximal. Math. Z. 137, 285–310 (1974). MR 0460712 18. A. Grothendieck, Éléments de géométrie algébrique. Inst. Hautes Études Sci. Publ. Math. (1960– 1967) 19. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203 (1964); ibid. (2) 79, 205–326 (1964). MR 0199184 20. H. Hironaka, Idealistic Exponents of Singularity, Algebraic Geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, MD, 1976), (Johns Hopkins Univ. Press, Baltimore, MD, 1977), pp. 52–125 MR 0498562 21. K. Kato, Logarithmic Structures of Fontaine-Illusie, Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988) (Johns Hopkins Univ. Press, Baltimore, MD, 1989), pp. 191–224. MR 1463703 (99b:14020) 22. G. Kempf, F.F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings. I, Lecture Notes in Mathematics, vol. 339 (Springer, Berlin, 1973). MR 0335518 (49 #299) 23. J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166 (Princeton University Press, Princeton, NJ, 2007). MR 2289519 (2008f:14026) 24. K. Kurano, K. Shimomoto, Ideal-adic completion of quasi-excellent rings (after Gabber). Kyoto J. Math. 61(3), 707–722 (2021). MR 4301055 25. M. McQuillan, Very functorial, very fast, and very easy resolution of singularities. Geom. Funct. Anal. 30(3), 858–909 (2020). MR 4135673 26. M. McQuillan, D. Panazzolo, Almost étale resolution of foliations. J. Differential Geom. 95(2), 279–319 (2013). MR 3128985 27. W. Nizioł, K-theory of log-schemes. I. Doc. Math. 13, 505–551 (2008). MR 2452875 28. D. Panazzolo, Resolution of singularities of real-analytic vector fields in dimension three. Acta Math. 197(2), 167–289 (2006). MR 2296055 29. M.H. Quek, D. Rydh, Weighted blow-ups, In preparation. https://people.kth.se/~dary/weightedblowups20220329.pdf 30. M.H. Quek, Logarithmic resolution via weighted toroidal blow-ups. Algebr. Geom. 9(3), 311– 363 (2022). MR 4436684 31. A.J. Schwartz, Functorial smoothing of morphisms in equal characteristic 0, Ph.D. thesis, Harvard University, 1992, p. 68. MR 2687722 32. M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero. Adv. Math. 219(2), 488–522 (2008). MR 2435647 33. M. Temkin, Stable modification of relative curves. J. Algebraic Geom. 19(4), 603–677 (2010). MR 2669727 34. M. Temkin, Absolute Desingularization in Characteristic Zero, Motivic Integration and Its Interactions with Model Theory and Non-Archimedean Geometry. Volume II, London Math. Soc. Lecture Note Ser., vol. 384 (Cambridge Univ. Press, Cambridge, 2011), pp. 213–250. MR 2905858 35. M. Temkin, Functorial desingularization of quasi-excellent schemes in characteristic zero: the nonembedded case. Duke Math. J. 161(11), 2207–2254 (2012). MR 2957701

218

M. Temkin

36. M. Temkin, Functorial desingularization over q: boundaries and the embedded case. Israel J. Math. 224(1), 455–504 (2018). MR 3799764 37. M. Temkin, Introduction to logarithmic geometry, in New Techniques in Resolution of Singularities (Springer, Cham, 2023), this volume 38. O. Villamayor, Constructiveness of Hironaka’s resolution. Ann. Sci. École Norm. Sup. (4) 22(1), 1–32 (1989). MR 985852 39. J. Włodarczyk, Simple Hironaka resolution in characteristic zero. J. Amer. Math. Soc. 18(4), 779–822 (2005) (electronic). MR 2163383 40. J. Włodarczyk, Functorial resolution by torus actions, in New Techniques in Resolution of Singularities (Springer, Cham, 2023), this volume

Weighted Resolution of Singularities. A Rees Algebra Approach Jarosław Włodarczyk

1

Introduction

Weighted blow-ups of varieties and schemes have already been present in birational geometry for some time. They found remarkable applications in the works of Varchenko, Reid, Kawamata, Artal-Bartolo, Martin-Morales, and Ortigas-Galindo [7,24,37,42]. They are also at the very center of the theory of toric varieties [16, 20, 33], and most of the toric resolution algorithms. One of the critical features of weighted blow-ups is that they adapt to the problems of computing various resolutions and resolution invariants surprisingly well, producing abelian quotient singularities. Quite recently, Panazzolo [34] used scheme theoretic weighted blow-ups to resolve foliations in dimension three. Subsequently, McQuillan and Panazzolo in [30] used stack-theoretic blow ups to further enhance the results. The latter work led to the paper [27], where stack-theoretic weighted blow-ups were used for embedded resolution. On the other hand, in the series of papers by Abramovich-Temkin-Włodarczyk [4–6] all kinds of stack-theoretic blow-ups were considered to simplify resolution procedure or to adapt the algorithm to the relative situation of morphisms. In particular, in [5, 6], we give a functorial logarithmic resolution for any dominant morphisms. In this case, the use of stack-theoretic blow-ups was dictated by the functoriality properties.

This research is supported by BSF grant 2014365. J. Włodarczyk () Department of Mathematics, Purdue University, West Lafayette, IN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. Abramovich et al., New Techniques in Resolution of Singularities, Oberwolfach Seminars 50, https://doi.org/10.1007/978-3-031-32115-3_6

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Moreover, in [4], independently of the work [27], we consider stack- theoretic weighted blow-ups to obtain a fast and efficient stack-theoretic embedded and non-embedded resolution of singularities. The algorithm is related to the Hironaka resolution method with smooth centers and should be regarded as its far-going simplification. The significant advantage of the weighted centers and their generalizations is that they naturally occur in the resolution method and, as such, give an immediate improvement of the singularities. This improvement is controlled by a simple geometric invariant related to the weighted normal cone. The weighted normal cone approximates the singularities for the resolution purpose, and the weights associated with the weighted cone form a natural invariant that improves after each blow-up. When using smooth centers, this is no longer the case. As can be easily seen in many examples, the functorial embedded resolution by smooth blow-ups, which relies solely on the local scheme structure, is impossible (see Example 1.5.) In other words, additional data on the smooth ambient variety need to be introduced to resolve singularities by the Hironaka method with smooth centers. The resolution process produces a simple normal crossings divisor (shortly SNC divisor) by accumulating exceptional divisors of the blow-ups. The SNC divisor is locally described by the product of local coordinates for a certain coordinate system. To keep the resolution algorithm running, we need a particular filtration on the smooth logarithmic structure given by an SNC divisor, commonly referred to as “history.” Even the logarithmic structure alone is insufficient to break up the infinite loop determined by the functoriality properties. The SNC structure that shall be carried in the algorithm creates additional constraints on the centers needing SNC with the SNC exceptional divisor. The filtration of the SNC divisor used in the process is the primary source of the technical difficulties of the algorithm. Smooth blow-ups rarely improve singularities. We can only see such an improvement in the process’s “history” or by introducing an additional datum. On one side, the resolution algorithm can be broken down into inductive steps on simple invariants. On the other hand, computing the main resolution invariant and its modifications uses many steps, is usually tiresome, and conceptually hard to grasp even in relatively simple situations. Thus the problem of the complexity of the Hironaka resolution lies in the nature of smooth blow-ups. One of the main reasons the smooth centers are being used in the embedded resolution is to keep the ambient variety smooth while resolving the singularities of the subvariety. This is also the main disadvantage of the ordinary weighted blow-ups of schemes, as they lead to singular ambient spaces. To employ weighted centers, one needs to redefine the operation of blow-ups and the notion of smoothness. One way to go is to replace the weighted blow-ups of varieties with stack-theoretic ones. Our smooth

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ambient variety, in that case, should be replaced by a smooth Deligne–Mumford stack. Locally such a stack can be represented by a smooth variety with finite diagonal group action. This is exactly done in papers [27] of McQuilan-Marzo, and [4] by AbramovichTemkin-Włodarczyk. In the subsequent paper, [2] of Abramovich-Quek, the definition is further extended to more general multiple weighted blow-ups with locally monomial centers. This gives a perspective of employing even more general centers while staying in the smooth category. The stack-theoretic approach brings many benefits to the resolution problem. The price we need to pay is more complex ambient objects, which are Deligne–Mumford stacks. In [49], the smooth ambient stacks were replaced by the smooth varieties with torus action. Consequently, the stack-theoretic weighted blow-ups are substituted by the cobordant weighted blow-ups. The additional advantage of this language is a valuable information carried by the torus action and simple transformation rules. The operation of cobordant blow-ups brings the action of .Gm and, more generally, k .T = Gm into play. In fact, .Gm -actions already play a vital role in birational geometry. In the work of Reid, Thaddeus, and many others (see [17, 37, 39–41]) a clear connection of torus actions with the Mori theory was established. In particular, Reid [37] emphasized the role of weighted blowings up and flips in birational geometry using .Gm -actions. This was also reflected in the proof of the weak factorization theorem, which relied on the notion of birational cobordism and a critical role of .Gm -action [3, 44, 45]. The formalism of the cobordant weighted blow-ups from [49] was further generalized in the subsequent paper [50]. The cobordant weighted blow-ups shall be thought of as the representation of the weighted blow-ups of schemes in terms of a torus action. This concept is extended in [50], where we show that any proper birational morphism of normal varieties can be represented by a simpler morphism of spaces with torus actions via a variant of Cox rings. In many cases, such a canonical presentation, called cobordization, gives a very convenient homogenous description of the original morphism. In the case of weighted blow-up of smooth schemes, the cobordization is simply the weighted cobordant blow-up from [49]. The cobordization of locally monomial ideals on smooth varieties gives the cobordant version of Abramovich-Quek multiple weighted blow-ups from [2].

1.1

Applications of Weighted and Generalized Blow-Ups

The weighted blow-up’s particular simplicity and flexibility make them a valuable tool for algorithmic computations and possible theoretical implementations. The method can be applied to many existing and potential embedded resolution algorithms, particularly those initially relying on smooth centers, regardless of characteristic.

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In particular, the proofs in [4] and [49] can be regarded as highly simplified versions of the strong Hironaka algorithm in characteristic zero. They also rely on the basic concepts of the standard Hironaka algorithm from [10, 18, 19, 23, 26, 43, 46], which are adapted to the new language. Although the resolution method mentioned above with weighted centers stems from the Hironaka approach it brings a substantial simplification. The difference between the algorithms can be seen in many examples where the strategy with smooth centers requires multiple steps, literally hundreds or thousands. In contrast, the weighted algorithm can be reduced to very few weighted blow-ups and thus is more suitable for direct computations. While the centers of the algorithm in [4] are written as .Q-ideals. the approach in [49] relies on rational Rees algebras. Moreover in the second paper, the SNC divisor (without filtration) is added to the structure of the ambient scheme to obtain the monomial principalization and the resolution with SNC exceptional divisors. The idea of using rational Rees algebras, introduced in the first version of [4], and used as a basic tool in [49] is to expand the original .Z≥0 -gradation of the Rees algebra associated with the blow-up and pass to a finer .Q-gradation in order obtain a much nicer representation of the center. In fact, the whole algorithm is run inside of the Rees algebra of the center, which itself is obtained by a simple recursive formula leading to fast computation process requiring minimal number of steps. With this approach uniqueness of the center is automatic, which is a novelty for the embedded resolution algorithms. Although the main principles of the algorithm stem from the Hironaka method they are reshaped in a new form with often different defining properties. The invariant used here is apriori characteristic free and can be considered in a more general context. It is defined in the language of rational Rees algebras, and can be easily computed in characteristic zero. On the other hand, when considering the invariant beyond the characteristic zero, one can see its pathological behavior (see Sect. 5.1). Nevertheless, it still has its applications in many particular situations (see [49]). Thus its role in the general resolution of singularities in nonzero characteristic remains unclear. The immediate consequence of using weighted blow-ups in positive characteristic is that it erases all types of problems associated with monomial divisorial structure and smooth centers. The first such examples were discovered by Moh and Abhyankar [1, 28]. Since then, they have been studied intensely, and some cases were even classified in the papers [11, 13–15, 21, 25, 29, 47]. One should mention, in particular, a recent spectacular example by Perlega in [22] expressing the infinite increase of so-called residual order. However, all these examples become of much lesser value in the view of the method of weighted blow-ups, where the invariants are defined without relevance to the divisorial structure and are meant to improve singularities directly without additional factoring of the accumulated monomials.

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As we already mentioned, there are still many pathologies in characteristic p, which do not refer to any divisorial structure. They were also studied in some of the earlier papers, with the most famous Narasinhman example [32]. A few of the problems associated with the main resolution invariant will be illustrated in Sect. 5.1. Nevertheless, the method of the weighted centers, a priori, eliminates many of the technical difficulties associated with positive characteristic and exposes the essential stumbling blocks for the resolution. Some of these pathologies are discussed in Sect. 5.1. Resolution algorithms with weighted centers as in [4, 27, 49] output varieties with abelian quotient singularities or corresponding tame Artin stack (with finite inertia) in the sense of .ℵ–Olsson–Vistoli. It can be directly resolved by canonical combinatorial methods in any characteristic as in [48], or in [12] by Bergh-Rydh.

1.2

Invariant and Centers

Perhaps the most striking feature of our resolution method is the simplicity of the resolution invariant (see Definition 4.5). If .I is an ideal on regular X with no divisorial structure and .p ∈ X is a point then we define the invariant in the language of the rational Rees algebras: invp (I) = max{(a1 , . . . , ak ) |

.

It ⊂ OX [x1 t 1/a1 , . . . , xk t 1/ak ]int },

where .x1 , . . . , xk is a certain partial coordinate system, .wA = lcm(a1 , . . . , ak ) is the smallest positive rational number which is a common integral multiple of .a1 , . . . , ak , and the integral closure is taken in .OX [t 1/wA ]. Note that we use here a modified version of the standard notion of lcm. The actual definition of the invariant in the case of a regular scheme with an SNC divisor is slightly different (see Definition 4.5). In the present definition we assign with the ideal .I weight 1 and place it in t-gradation, while giving weight .1/ai to .xi . As we mentioned before the rational Rees algebra were already considered in the resolution context in the first version of [4] although the algorithm there is run in the language of .Q-ideals discussed in Sect. 2.1. The notion of the invariant .invp (I) is very geometric and is linked to the weighted normal cone. It naturally generalizes the order ordp (I) = max{a ∈ N | I ⊂ map }

.

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which can be written in this language as ordp (I) = max{a1 ∈ Q |

.

It ⊂ OX [mp t 1/a1 ]}.

(See Lemma 4.15). Our invariant .inv can be linked to a part of the invariant used in the simplified Hironaka method with smooth centers. The latter is however way more involved and is determined in the inductive procedure which relies on an additional datum. Example 1.1 Let .f = x1a1 + . . . + xkak ∈ K[x1 , . . . , xk ], where a1 ≤ . . . ≤ ak .

.

If the characteristic of the base field K is zero, the subvariety .V (f ) ⊂ Ak admits an isolated singularity at the origin 0. In this case we have f t ⊂ OX [(x1 t 1/a1 , . . . , xk t 1/ak ]int ,

.

and the invariant associated with the origin can be easily computed as inv0 (f ) = (a1 , . . . , ak ).

.

(See Example 4.86). The function f becomes quasi-homogeneous for coordinates .xi with the weights .1/ai , while the weight of f is simply 1. These weights determine the associated weighted tangent cone .Cw (f ) defined by the quasihomogeneous initial part of f . In our case, .Cw (f ) can be identified with .V (f ). Equivalently the numbers .ai could be thought of as the “orders” of the ideal .(f ) in the direction of .xi . The main resolution invariant .inv0 (I) = (a1 , . . . , ak ) is tied to the weighted center described as the extended rational Rees algebra A = OX [t −1/wA , x1 t 1/a1 , . . . , xk t 1/ak ]

.

(see Sections 4.1.3) which defines the cobordant blow-up as B = SpecX (OX [t −1/wA , x1 t 1/a1 , . . . , xk t 1/ak ]

.

The numbers .1/ai , and .−1/wA can be rescaled to the integral weights .wi = wA /ai of .xi , and the formula for B can be conveniently written using integral weights in the equivalent form: B = SpecX (OX [t −1 , x1 t w1 , . . . , xk t wk ])

.

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Similarly to the ordinary blow-up with a smooth center, the cobordant weighted blowup .B → X takes the center .A into the invertible sheaf corresponding to the exceptional locally irreducible divisor associated with Rees algebra generated by the principal ideal in the gradation t.

1.3

Main Results

The following results summarize the resolution method by stack-theoretic weighted blowups from [4] and their cobordant version with SNC divisors in [49]. One shall mention that the weighted resolution with cobordant blow-ups of varieties can be easily converted into the stack-theoretic resolution.

1.3.1 Functorial Principalization Theorem 1.2 Let X be a smooth variety over a field K of characteristic zero, and E be a simple normal crossing (SNC) divisor and I be an ideal sheaf on X. There exists a canonical principalization of I, that is a sequence of cobordant blow-ups (Definition 2.9) at smooth weighted centers X = X0 ← X1 ← . . . ← Xk = X ,

.

(∗)

such that: (1) The torus Ti = Gim acts on Xi , with finite stabilizers, where T0 = 1, Ti+1 = Ti × Gm , ans T := Tk so that the geometric quotient (space of orbits) Xi /Ti exists. (2) Set I0 := I. For i ≥ 1, the ideals Ii ⊂ OXi , defined as Ii := OXi · Ii−1 are Ti -stable. (3) Set E0 := E, and let Ei , for i ≥ 1, be the total transform of Ei−1 . The divisors Ei on Xi are Ti -stable and have SNC. (4) The smooth weighted centers Ji of the cobordant blow-ups Xi ← Xi+1 are compatible with Ei (Definition 4.2), and are Ti -stable. Moreover V (Ji ) ⊆ V (Ii ). (5) OX · I = ID := OX (−D ) is the ideal of an SNC divisor D whose components are the strict transforms of the components of the total transform E of E. Moreover there is a T -equivariant isomorphism over X  V (I): X  D (X  V (I)) × T .

.

(6) The sequence (*) defines a sequence of weighted blow-ups on the induced geometric quotients X = X/T0 ← X1 /T1 ← . . . ← Xk /Tk = X /T , such that OX /T · I = ID is the ideal of a locally principal divisor D := D /T ⊂ X /T , where D is a T -stable SNC divisor on X , and OX · ID = ID .

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(7) The sequence (*) defines a sequence of stack-theoretic weighted blow-ups on the induced smooth stack-theoretic quotients X = X0 = [X/T0 ] ← [X1 /T1 ] ← . . . ← [Xk /Tk ] = [X /T ],

.

such that O[X /T ] · I is the ideal of an SNC divisor on [X /T ]. (8) The sequence (*) is functorial for smooth morphisms, field extensions, and group actions preserving (I, E).

1.3.2 Embedded Desingularization Theorem 1.3 Let Y be a reduced closed subvariety of a smooth variety X over a field K of characteristic zero. There exists a sequence of cobordant blow-ups (Definition 2.9) at smooth weighted centers X = X0 ← X1 ← . . . ← Xk = X

.

(∗)

and the induced sequence of the strict transforms such that Y = Y0 ← Y1 ← . . . ← Yk = Y . (1) The torus Ti = Gim acts on Xi , where Ti+1 = Ti × Gm , T := Tk so that the geometric quotient (space of orbits) Xi /Ti exist. (2) The closed subvarieties Yi ⊂ Xi are Ti -stable. In particular, they admit geometric quotients Yi /Ti . (3) The smooth weighted centers of the cobordant blow-ups Xi ← Xi+1 are Ti -invariant and are contained in Yi . (4) Y is a smooth T -stable subvariety of X . (5) The sequence (*) defines the sequence of the weighted blow-ups on the induced geometric quotients X = X/T0 ← X1 /T1 ← . . . ← Xk /Tk = X /T , such that Y /T ⊂ X /T admit abelian quotient singularities, (6) The sequence (*) defines the sequence of the stack- theoretic weighted blow-ups on the induced smooth stack theoretic quotients X = X0 = [X/T0 ] ← [X1 /T1 ] ← . . . ← [Xk /Tk ] = [X /T ],

.

such that [Y /T ] ⊂ [X /T ] is a smooth substack. (7) The sequence (*) is functorial for smooth morphisms, group actions, and field extensions.

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1.3.3 Nonembedded Desingularization Theorem 1.4 Let Y be a reduced variety of finite type over a field K of characteristic zero. There exists a sequence of cobordant blow-ups at smooth weighted centers: Y = Y0 ← Y 1 ← . . . ← Yk = Y

.

(∗∗)

such that (1) Y is a smooth variety. (2) The torus Ti = Gim acts on Yi , where Ti+1 = Ti × Gm , T := Tk , so that the geometric quotient Yi /Ti exist. (3) The cobordant blow-ups Yi ← Yi+1 are Ti -invariant. (4) The sequence (**) defines the sequence of the weighted blow-ups on the induced geometric quotients Y = Y /T0 ← Y1 /T1 ← . . . ← Yk /Tk = Y /T , such that Y /T has an abelian quotient singularities. (5) The sequence (**) defines the sequence of the stack-theoretic weighted blow-ups on the induced stack-theoretic quotients Y = Y0 = [Y /T0 ] ← [Y1 /T1 ] ← . . . ← [Yk /Tk ] = [Y /T ],

.

such that [Y /T ] is a smooth stack. (6) The sequence (**) is functorial for smooth morphisms, group actions, and field extensions.

1.4

Motivating Examples

There are numerous examples where one can illustrate the advantage of the method with the weighted centers over the ordinary smooth centers. Example 1.5 ([4]) Consider the equation f = x 2 − y1 y2 y3 ∈ K[x, y1 , y2 , y3 ].

.

The singular locus of .V (f ) ⊂ A4 consists of the three lines, .x = yi = yj = 0 where .(i, j ) is defined by a subset of two elements out of .{1, 2, 3}. A symmetry group of automorphisms acting on the hypersurface and permuting coordinates .y1 , y2 , y3 . This example shows that the functorial resolution method by smooth blow-ups, which relies only on the local description of singularities with no additional structure, does not exist. In the resolution process, we usually consider the functoriality properties for étale or smooth morphisms, but it suffices to assume functoriality for open embeddings.

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Consequently, let us assume that the open embeddings should preserve the resolution procedure. The only possible smooth center of the blow-up which is preserved by the automorphism group is the origin. In the chart associated with .y1 , we have the coordinate change x = x y1 , y1 = y1 , y2 = y2 y1 , y3 = y3 y1 ,

.

where .y1 is the equation of the exceptional divisor. It transforms .x 2 − y1 y2 y3 into (y )2 ((x )2 − (y1 y2 y3 )).

.

After clearing up the exceptional divisor we obtain the same equation in all three charts associated with .yi . Thus, the situation worsens as we replace the origin with the three locally isomorphic singularities on the strict transform of .V (f ). By functoriality, all those isomorphic isolated singularities need to be blown up in the next step. This leads to an infinite loop of blow-ups that multiply the number of identical singularities. The Hironaka method, which breaks up this loop, relies on the additional information encoded in the divisorial structure. In our case, one of the coordinates .yi becomes divisorial at every new singular point, and its role in the algorithm changes. Still, the method repeats such blow-ups of isomorphic singularities a few more times, multiplying each time its number and producing the relevant exponential number of identical singularities in the numerous charts. The changes are, however, reflected in the invariant, which drops after each blow-up due to the different roles of the exceptional divisors for the associated filtration. Only after this preliminary stage of producing several dozen identical singularities associated with a few distinct values of the invariant do we start seeing a more visible improvement of the singularities. On the other hand, the method of weighted blow-up, which is still functorial, requires just two blow-ups corresponding to the origin and the reducible center of three smooth lines of the singular locus. The latter becomes smooth after the first blow-up. The first cobordant blow-up of the center Aext = OX [t −1/6 , xt 1/3 , y1 t 1/2 , y2 t 1/2 , y3 t 1/2 ])

.

is given by B = Spec(OX [t −1 , xt 3 , y1 t 2 , y2 t 2 , y3 t 2 ] = Spec(K[t −1 , xt 3 , y1 t 2 , y2 t 2 , y3 t 2 ])

.

and

.

B+ = B  V (xt 3 , y1 t 2 , y2 t 2 , y3 t 2 ).

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We write the transform of .f = x 2 − y1 y2 y3 in the new variables .x := xt 3 , y1 := y1 t 2 , y2 := y2 t 2 , y3 := y3 t 2 as t −6 ((x )2 − y1 y2 y3 )

.

So, after clearing the exceptional divisor .t −1 in .B X × A1 we get the exactly the same equation on B: (x )2 − y1 y2 y3 = 0.

.

Passing to the cobordant blow-up .B+ = B  V (x , y1 , y2 , y3 ) we eliminate the worst singularity component, which is the product of the origin and .A1 . Consequently, the three components of the singular locus (arising from the three lines) are now smooth and disjoint. Thus one can take a reducible smooth center with weights one and apply the corresponding cobordant blow-up. This time .B+ is a locally trivial .K ∗ bundle over the standard blow-up. So on the level of geometric quotients, both operations: The cobordant blow-up of a smooth center and the ordinary blow-ups of the same center, coincide. The second smooth cobordant blow-up finally resolves the singularity. Example 1.6 ([49]) The difference between the weighted and the standard smooth centers is even more visible in Example 1.1, given by the equation f = x1a1 + . . . + xkak

.

in .Ak . The smooth blow-ups here typically worsen singularities. For instance, the blow-up at the origin at the .xk -chart is defined by .x1 = x1 xk transforms the equation into k−1 x1a1 + x2a2 xka2 −a1 + . . . + xk−1 xk k−1

a

.

a

−a1

+ xkak −a1

Then numerous steps are needed to resolve the singularities of that form. The procedure is tedious, and the progress is visible only on the level of the involved inductive structure, which relies on the careful bookkeeping of the data. The situation is dramatically different when working with the weighted centers. In this case we consider the weighted cobordant blow-up of the center .Aext = 1/a 1/a Spec(OX [t −1/wA , x1 1 , . . . , xk k ]). Putting .wi = w/ai we get B = Spec(K[t −1 , x1 t w1 , . . . , xk t wk ] X × A1

.

and .B+ = B  V (x1 t w1 , . . . , xk t wk ).

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Writing our equation on B in the new coordinates .t −1 , x1 := t w1 x1 , . . . , xk := t wk xk , we get the formula (x1a1 + . . . + xkak ) = t −wA ((x1 )a1 + . . . + (xk )ak ) = 0

.

We obtain exactly the same equation after clearing the maximal power of the exceptional divisor .t −wA . Consequently, when passing to .B+ = B  V (x1 , . . . , xk ), we eliminate the singular locus of the strict transform of .V (f ) resolving the singularities in a single step. Again, the whole operation looks surprisingly simple at the first glance. However, we increased the dimension and introduced a nontrivial action of .Gm on .B+ to accommodate the weights of the center.

2

Cobordant and Stack-Theoretic Blow-Ups

2.1

Q-Ideals

In [4], we introduced the notion of .Q-ideals to consider rational powers of ideals on a variety. The concept of rational powers of the elements is present in many resolution algorithms. The rational powers of monomials in divisorial unknowns allowing for much easier computations were used already in Bierstone-Milman approach [9, 10]. The concept is used broadly in toric geometry [16, 20, 33]. The different notions of rational powers of monomials were used in the recent papers [4], in [6], and in [35] in the context of the logarithmic and the relative logarithmic resolutions. Finally, in [4], the concept of .Q-ideals is used to describe the centers of the weighted blow-ups. There are several different approaches to the .Q-ideals. An extensive treatment of this notion is given in the papers [35, 36], where the language of Rees algebra is used. In [49] the notion of .Q-ideals was replaced by a more general notion of the rational Rees algebras considered in this paper. The following definition is equivalent to a particular case of a more general notion introduced in [4]. Definition 2.1 For an ideal .I on X and .n ∈ N consider the formal expression .I 1/n . By the 1/n , where .Q-ideals on a normal X, we mean the equivalence classes of the expressions .I 1/n and .J 1/m are equivalent if the ideals .I m = I mn/n .I is an ideal on X. We say that .I n mn/m and .J = J have the same integral closure.

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Any such an expression .J = I 1/n corresponds to the integral closure AJ := (OX [It n ])int

.

of the graded Rees algebra OX [It n ] = OX ⊕ It n ⊕ I 2 t 2n ⊕ . . .

.

in .OX [t]. Conversely any integrally closed Rees algebra OX ⊕ I1 t ⊕ . . . ⊕ In t n ⊕ . . .

.

can be written as the integral closure of .OX [It n ]int for a certain large n, and .I = In . This leads to the following observation made by Quek: Definition 2.2 ([35]) There is a bijective correspondence between the .Q-ideals .J on a normal scheme X and the integrally closed algebras on X, given by J → AJ .

.

Under this correspondence the ordinary ideal .I corresponds to .OX [It]int .  a /n Observe that the formal sum . ki=1 Ii i i , for the ideals .Ii on X, and .ai , ni ∈ N defines a .Q-ideal on X corresponding to the Rees algebra an

OX [I1a1 t n1 , . . . , Ik k t nk ]int .

.

In this language we can write on .Q-ideals:  .

1/n

Ii

=(



Ii )1/n

I 1/n = (I 1/mn )m .

and

This determines the operations of addition and multiplication on the .Q-ideals on X . For  1/ni  1/nj Ii and .J2 := Jj the ideal .Q-ideals on X written in a general form as .J1 := their sum is given by J1 + J2 =

.



1/ni

Ii

+



1/nj

Jj

and their product .J1 ·J2

=

 n  1  1/n  1/nj  1/n  nj n n 1n 1/nj n n n j Ii i · Jj = Ii i ·Jj = (Ii ·Jj i ) i j = (Ii ·Jj i ) i j i,j

i,j

i,j

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are also .Q-ideals on X. The algebra .AJ1 +J2 is simply the integral closure of subalgebra of .OX [t] generated by .AJ1 and .AJ2 , while .AJ1 ·J2 is the integral closure of the algebra generated by the products of the gradations .(AJ1 )k · (AJ1 )k t k of .AJ1 and .AJ2 . These operations extend the standard notion of the sum and the product of ideals.

2.1.1 Graded Q-Ideals With any .Q-ideal .J = I 1/n on X we can associate the (ordinary) ideal .(J )X on X defined as JX := {f ∈ OX | f n ∈ I int }.

.

Thus .(J )X can be interpreted as the first gradation .A1 of If .J = I 1/n is a .Q-ideal on X then (J m )X = (f ∈ OX | f n ∈ (J m )int )

.

is nothing but the m-th gradation .Am of .A = AJ . This follows from the fact that mn gradation .Amn of .AJ is .(I m )int . Thus the graded algebra .AJ associated with the .Q-ideal .J can be written as the Rees algebra AJ = (OX [J t])X :=

.



(J i )X t i

Here we think of OX [J t] :=

.



J iti

as the graded algebra of .Q-ideals on X generated by its first gradation .J t.

2.1.2 Vanishing Locus Given a .Q-ideal .J = I 1/n on X one can associate with it the vanishing locus .V (I) on X to be V (J ) = V (I 1/n ) = V (I) = V ((AJ )>0 ).

.

2.1.3 Blow-Ups of Q-Ideals Let .I be an ideal on a normal variety X. Consider the normalized blow-up Y = P roj (OX [It]int ) → X

.

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233

of .I. Then the inverse image .OY · I of .I defines a Cartier divisor D corresponding to the invertible relatively very ample sheaf .OY (1) = OY (−D), while the .Q-ideal .OY · I 1/n corresponds to the .Q-Cartier divisor .(1/n) · D. On the other hand the normalized blow-up of .Q-ideal .J = I 1/n can be written as Y = P roj (OX [It n ]int ) = P roj ((OX [J t])X ) → X.

.

It is isomorphic to the normalized blow-up of .I but admits a different presentation. Note that the inverse image .I · OX [It n ]int of .I corresponds to the twisted sheaf .OY (n).

2.1.4 Smooth Weighted Centers In the resolution algorithm [4] we considered the valuative .Q-ideals on a smooth variety X, called centers which can be locally presented in the form .J = (ua11 , . . . , uakk ) where .u1 , . . . , uk is a set of local parameters where .a1 ≤ . . . ≤ ak are positive rational numbers. Equivalently we can rewrite the center as nak 1/n 1 J = (ua11 , . . . , uak ) = (una ) 1 ,...,u

.

for a sufficiently divisible n with the integral powers .nak . The induced ideal on X will be denoted by .JX = (ua11 , . . . , uakk )X . One can describe these ideals in a few different ways. By a simple direct computation we have 1/w

1/w

Lemma 2.3 ([4, 49]) Let .J = (u1 1 , . . . , uk k ) be a center on a regular X, where .wi ∈ N, and such that that .V (u1 , , . . . , uk ) is irreducible. Let .νJ be a monomial valuation such that .νJ (ui ) = wi , and for any .a > 0, .IνJ ,a := {f ∈ OX | νJ (f ) ≥ a} is equal to α α .(u | νJ (u ) ≥ a). Then 1/w1

((u1

.

1/wk a

, . . . , uk

) )X = IνJ ,a = {f ∈ OX | νJ (f ) ≥ a} =

(ub11 · . . . · ubkk | bi ∈ N, b1 w1 + . . . bk wk ≥ a).

.

In particular ((ua11 , . . . , uakk ))X = (ub11 · . . . · ubkk | bi ∈ N, b1 /a1 + . . . + bk /ak ≥ 1).

.

♣

.

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Consequently, we have Lemma 2.4 ([4, 35, 49]) Let .(u1 , , . . . , uk ) be a partial system of local parameters on regular X, such that that .V (u1 , , . . . , uk ) is irreducible. Let .w1 ≥ w2 ≥ . . . wk be positive integers. There is a natural bijective correspondence between 1/w1

(1) valuative ideals .J = (u1 (2) graded algebras

1/wk

, . . . , uk

).

AJ = OX [u1 t a1 , . . . , uk t ak | a1 ≤ w1 ] = OX [t −1 , u1 t a1 , . . . , uk t ak ]≥0 .

.

(3) monomial valuation .νJ , such that .νJ (ui ) = wi , and .IνJ ,a := {f ∈ OX | νJ (f ) ≥ a} is equal to .(uα | νJ (uα ) ≥ a). Moreover we get AJ =



.

IνJ ,a t a = (OX [J · t])X .

a∈Z≥0

♣

.

2.1.5 Stack-Theoretic Blow-Ups of Q-Ideals Let X be a variety over a base field K. In [4], the stack-theoretic blow-up of any valuative ideal .I is defined to be the stack-theoretic quotient of the graded algebra (using our notation): [(Spec(OX [It])X  V (It)/Gm ],

.

with the standard action of the multiplicative group .Gm := Spec K[t, t −1 ]. This formula is a natural extension of the standard formula for the blow-up of the ideal .I: Proj (

∞ 

.

I i t i ).

i=0

2.1.6 Stack-Theoretic Weighted Blow-Ups 1/w 1/w In particular, the stack-theoretic weighted blow-up of .J = (u1 1 , . . . , uk k ) is defined to be the stack-theoretic quotient of the graded algebra .

Spec(OX [J t])X = [(Spec(OX [t w1 x1 , . . . , t wk xk ]int )  V (t w1 x1 , . . . , t wk xk ))/Gm ].

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Observe that the space .

Spec(OY [t w1 x1 , . . . , t wk xk ]int ) = Spec(OY [t a1 x1 , . . . , t ak xk ]0≤ai ≤w1 ) = Spec(AJ )

is singular even for the standard smooth blow-up of the center .(x1 , . . . , xk ) with weights wi = 1. To represent the weighted blow-ups by the torus quotients of smooth spaces, we use the concept of birational cobordisms.

.

2.1.7 Good and Geometric Quotient We shall consider here a relatively affine action of T = Spec(Z[t1 , t1−1 , . . . , tk , tk−1 ])

.

on a scheme X over .Z. By the good quotient (or GIT quotient) we mean an affine invariant morphism .π : X → Y = X//T such that the induced morphism of the sheaves .OY → π∗ (OX ) defines the isomorphism onto the subsheaf of invariants .OY π∗ (OX )T . Then .π : X → Y = X/T will be called the geometric quotient if additionally every fiber .Xy of .π over s geometric point .y : Spec(K) → Y defines a single orbit of the action of .TK = T ×Z Spec(K) = Spec(K[t1 , t1−1 , . . . , tk , tk−1 ]) on .Xy .

2.1.8 Birational Cobordisms Consider the action of .T = Gm = Spec Z[t, t −1 ] on a scheme X. Definition 2.5 For any point .x ∈ X we that .limt→0 (tx) exists, (respectively .limt→∞ (tx) exists) if there is an open neighborhood U of x such that the morphism .Gm × U → X extends to the .Spec(Z[t]) × U → X (respectively to .Spec(Z[t −1 ]) × U → X). Remark 2.6 This definition differs slightly from the standard definition over an algebraically closed field K. Definition 2.7 ([44]) By a birational cobordism representing a proper birational map .φ : X1  X2 we mean a normal scheme B with an action of .T = Gm such that the sets .

B− := {x ∈ B : limt→0 t (x) does not exist in B} and B+ := {x ∈ B : limt→∞ t (x) does not exist in B}

are nonempty Zariski open subsets of B , and the geometric quotients .α1 : B+ /T X1 , and .α2 : B− /T X2 exists, with the natural birational map .ψ : B+ /T → B− /T defined by open inclusions map .(B− ∩ B+ )/T → B ± −/T , such that φ ◦ α1 = α2 ◦ ψ.

.

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The following example motivates the definition of a cobordant blow-up as the one representing weighted blow-ups in terms of the quotient of smooth space. Example 2.8 ([44]) Let T act on .B = An+1 = Spec(K[x0 , . . . , xn ]) by t (x0 , x1 , . . . , xn ) = (t −1 x0 , t w1 x1 , . . . , t wk xk ),

.

where .w1 ≥ w2 ≥ . . . ≥ wk > 0. Then .B− = B  V (x0 ), .B+ = B  V (x1 , . . . , xn ). The morphism B+ /T → B− /T = B//T = Spec(K[x0 , . . . , xk ])T = Spec(K[u1 , . . . , uk ]),

.

1/w

1/w

where .ui := xi x0wi is exactly the weighted blow-up at .J = (u1 1 , . . . , uk k ). This can be seen immediately when interpreting the morphism in the toric language as in Example 2.13. One can describe the morphism .B+ /T → B− /T = B//T explicitly. Set .t := x0−1 . Then we can write O(B) = K[x0 , x1 , . . . , xk ] = K[t −1 , t w1 u1 , . . . , t wk uk ].

.

So B+ /T = Spec(K[t −1 , t w1 u1 , . . . , t wk uk ]  V (t w1 u1 , . . . , t wk uk ) =

.

P roj (K[t −1 , t w1 u1 , . . . , t wk uk ]) = P roj (AJ ) → B− /T = Spec(K[u1 , . . . , uk ]). The first equality follows from the fact that .AJ is exactly the nonnegative part of the algebra .K[t −1 , t w1 u1 , . . . , t wk uk ], and in such a case Proj for both algebras are the same.

2.1.9 Cobordant Blow-Ups Let .T = Gm = Spec(Z[t, t −1 ]). Definition 2.9 ([49]) Let X be a regular scheme. A trivial cobordant blow-up of X is the product .X × T . 1/w 1/w Let .J be a center locally of the form .J = (x1 1 , . . . , xk k ) for a partial system of local coordinates .(x1 , . . . , xk ) scheme X. Let .AJ = (OX [J ·t])X be its associated algebra. By the full cobordant blow-up of X at a center .J we mean the T -invariant morphism .σ : B → X defined by the birational cobordism BJ = B := SpecX (AJ [t −1 ]) = SpecX (OX [t −1 , t w1 x1 , . . . , t wk xk ]).

.

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The upper and lower boundaries are equal respectively to B+ = B  V (t w1 x1 , . . . , t wk xk )

.

B− = B  V (t −1 ) = Spec OX [t, t −1 ].

.

By the cobordant blow-up we mean the T -invariant morphism .B+ → X. Here .B− → X is the trivial cobordant blow-up. We shall call the closed subscheme V = Vert(B) := V (t w1 x1 , . . . , t wk xk ) = B  B+

.

the vertex of B (per analogy to the vertex of the affine cone over a projective scheme.) Equivalently one can write B introducing a new variable .s = t −1 as B := SpecX (OX [s, s −w1 x1 , . . . , s −wk xk ]).

.

Remark 2.10 The algebras .OX [t −1 , t w1 x1 , . . . , t wk xk ] were independently discovered in the context of weighted stack-theoretic blow-ups by Rydh and Quek in [36]. Remark 2.11 (1) The geometric quotient .B+ /Gm → X can be naturally identified with the usual weighted blow-up. B+ /Gm (Proj (OX [t w1 x1 , . . . , t wk xk ]))nor → X = B//Gm = B− /Gm .

.

On the other hand the stack-theoretic quotient .[B+ /Gm ] → X coincides with the definition considered in [4] . (2) Similar to the standard blow-up the cobordant blow-up is trivial over the complement of the geometric locus of the center .V (J ) = V (x1 . . . , xk ). This allows to extend the definition to reducible centers. (3) By Lemma 2.4, the cobordant blow-up is uniquely determined by the center .J and is independent of choices of coordinates. (4) The vertex .Vert(B) plays important role in the resolution process. It is usually the locus of the points on B having the worst singularity, which are same as the singularities along the center .J , and are removed when passing to .B+ .

2.1.10 Exceptional Divisor 1 w

1 w

Lemma 2.12 The cobordant blow-up B+ transforms the center J = (x1 1 , . . . , xk k ) into the (ordinary) ideal of the exceptional divisor E on B+ generated by a global invariant

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parameter t −1 : IE := J · OB+ = t −1 OB+

.

Proof We have J · OB+ = (J · t)t −1 · OB+

.

Note however (J · t) = ((x1 t w1 )1/w1 , . . . , (xk t wk )1/wk ) = OB+ , is a trivial Q-ideal on ♣ B+ .

2.1.11 Cobordant Blow-Up: Local Equations Let .Gm = Spec(K[t, t −1 ]). Unlike the standard smooth blow-up, the full cobordant blow1/w 1/w up .B → X at .(x1 1 , . . . , xn n ) can be described by a single chart with the following coordinates • .t −1 is the inverse of the coordinate t representing the action of torus .Gm . • .xi = xi · t wi for .1 ≤ i ≤ k, and • .xj = xj for .j > k. The cobordant blow-up .B+ = B  V (x1 , . . . , xk ) can be covered by the open subsets .(B+ )x = B  V (x ), associated with .x producing several “charts” similarly to the i i i standard blow-up.

2.1.12 Stack-Theoretic Blow-Ups and Local Equations We describe the stack-theoretic weighted blow-up Y = [B+ /Gm ] → X

.

of .(x1 1 , . . . , xk k ) on X in the chart .Bx1 ⊂ B+ , where .x1 = x1 t w1 = 0. Consider the section .S1 ⊂ Bx1 defined by the equation .x1 = 1. Let .μw1 be a finite group generated by the .w1 -primitive root of unity .ξw1 . The group .μw1 is the stabilizer .Γp of the .Gm -action form all the points p in the section .S1 . Now, at least in characteristic zero, the stack theoretic quotient .[Bx1 /Gm ], can be reduced to the finite quotient .[S1 /μw1 ]. It follows from Sect. 2.1.10 that the exceptional 1/w 1/w divisor of the cobordant blow-up of .J = (x1 1 , . . . , xk k ) on .Bx1 is given by 1/w

1/w

OBx · J = t −1 · OBx .

.

1

1

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Consequently we have OS1 · J = OS1 · (t −1 )|S1

.

Thus the exceptional divisor of .[B+ /Gm ] → X in the chart .[Bx1 /Gm ] = [S1 /μw1 ] is −1 of the exceptional divisor .t −1 of .B+ → X. represented on .S1 by the restriction .y := t|S 1 Let us identify .xi with their restrictions to .S1 . Then

y w1 = (t −w1 )|S1 = (t −w1 · (x1 t w1 ))|S1 = x1

.

We also put yi := xi|S = (xi t wi )|S1 = xi y −wi 1

.

for .2 ≤ i ≤ k, and yi := xi|S = (xi )|S1 = xi 1

.

for .k + 1 ≤ i ≤ n. We obtain a coordinate system .(y, y2 , . . . , yn ) on the smooth .S1 ⊂ Bx1 t w1 which comes from the restriction of the partial coordinate system .(t −1 , x2 , . . . , xn ) on .Bx1 to .S1 , while the restriction .x1|S ≡ 1. 1 This gives us local formulas as in [4]: • .x1 = y w1 , • .yi = xi /y wi for .2 ≤ i ≤ k, and • .yj = xj for .j > k. The induced action of the isotropy group .μw1 ⊂ Gm on .S1 is given by (y, y2 , . . . , yn )

.

→

2 y , . . . , ξ −wk y , y (ξw1 y, ξw−w 2 k k+1 , . . . , yn ). w1 1

It is étale locally isomorphic to [Spec K[y, y2 , . . . , yn ]/μw1 ].

.

We obtain the stack-theoretic modification .π : Y = [B+ /Gm ] → X with a smooth Y . Its coarse space .Y coarse = B+ /Gm → X is the classical (singular) weighted blowing up as can be seen in Examples 2.13, and 2.8. On the open subset .U := X  V (J ), we have the identification for the cobordant blow-up: −1 πB+ (U ) = U × Gm ,

.

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which determines the isomorphism π −1 (U ) = [(U × Gm )/Gm ] U

.

and shows that the induced morphism of stacks .π : Y = [B+ /Gm ] → X is indeed a birational transformation of stacks and does not modify the complement of .V (J ).

2.1.13 Exceptional Divisor of the Stack Theoretic Weighted Blow-Up 1/w 1/w It follows from the above that for the stack theoretic blow-up of .J = (x1 1 , . . . , xk k ) the exceptional divisor D is given by the inverse image O[B+ /Gm ] · J = O[B+ /Gm ] (−D).

.

This can also be verified directly by the equalities of .Q-ideals in the charts: 1/w1

(x1

.

1/wk

, . . . , xk

1/w2

) · OY = (y, yx2

1/wk

, . . . , yxk

) = (y),

where .(y) = O[Bx /Gm ] (−D) describes the exceptional divisor and defines an invertible i sheaf on Y .

2.1.14 Cobordant Blow-Ups of Toric Varieties Example 2.13 ([44, 49]) We shall revisit Example 2.8 in the toric language. Let Xσ = Spec K[x1 , . . . , xn ] be the affine toric variety associated with the cone σ = Rn≥0 ⊂ NR := Rn . The full cobordant blow-up B = Xτ → Xσ of the Q- ideal J is a morphism of toric varieties corresponding to the inclusion of algebras: K[x1 , . . . , xn ] ⊆ K[t −1 , x1 t w1 , . . . , xn t wn ]

.

The standard weighted blow-up at J corresponds to a star subdivision of σ with the center v := w1 e1 + . . . + wn en . The full cobordant blow-up B is described by the regular cone τ := e1 , . . . , en , v + en+1  corresponding to K[t −1 , x1 t w1 , . . . , xn t wn ] with the natural projection onto σ = e1 , . . . , en  along the vector en+1 . The cone τ is a cobordism in the sense of Morelli [31]. The upper boundary τ+ of τ has maximal faces e1 , . . . , ei∨ , . . . , en , v + en+1  which are “visible from above” and which project exactly to the star subdivision of σ at v. It corresponds to the upper boundary B+ The lower boundary τ− is isomorphic to σ which is a unique cone “visible from below” and corresponds to B− = X × Gm . The projection π(τ+ ) corresponds to the quotient B+ /Gm , while σ = π(τ− ) describes B− /Gm X. The induced subdivision π(τ+ ) of σ is the star subdivision at v which is nothing but a weighted blow-up B+ /Gm → X = B− /Gm .

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2.2

241

Stack-Theoretic Weighted Blow-Ups vs Cobordant Blow-Ups

2.2.1 Rees Extended Algebras One can replace our original definition [(Spec(OX [It])int X )  V (It))/Gm ],

.

of stack-theoretic blow-up of a .Q-ideal .I on a variety X by the formula [(Spec(OX [t −1 , It])X )  V (It))/Gm ].

.

This approach can be linked to extended Rees algebra of an ideal. Typically the blow-up Y → X of an ideal .I is represented by the graded algebra

.

Y := Proj(OX [It]) = Proj(



I n t n ),

.

n∈N∪{0}

where t is a dummy unknown. However, Rees suggested a modification of the graded algebra by adjoining the additional generator .t −1 to obtain OX [t −1 , It] = OX t −1 ⊕



.

I nt n.

n∈N∪{0}

The generator .t −1 naturally corresponds to the twisting Serre .OY -module .OY (1) associated with the shifted algebra (OX [It])(1) = t −1 · OX [It].

.

On the other hand up to the negative gradations, we have (OX [It])(1) =



.

I n t n−1 = I · OX [It],

n∈N∪{0}

Consequently the inverse image I · OY = OY (−E)

.

of the blow-up of the ideal .I can be identified with relatively very ample divisor .−E on Y , corresponding to .OY (1). Both presentations of the exceptional divisors are linked by the

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relation t −1 · It = I,

.

with invertible .It. In the case of the .Q-ideal .J the graded algebra .OX [J t])int X is not generated by the first gradation .JX t and consequently .JX no longer generates .OY (1). By adjoining .t −1 we add the graded generator for the exceptional divisor .E = div(t −1 ), and of the Serre twisting sheaf .OY (1). The generator .t −1 is not locally represented by any ideal on the blow-up .Y → X of .J . However, it generates an exceptional divisor on the generalized cobordant blow-up B = Spec(OX [t −1 , J t])X ,

.

B+ = B  V (J t),

and on its stack-theoretic quotient defining the stack-theoretic blow-up.

2.2.2 Cobordant Blow-Ups The approach with extended algebras gives us nice regular representation of weighted 1/w 1/w blow-ups. If .J is a weighted center which is locally of the form .J = (u1 1 , . . . , uk k ) then by Lemma 2.4 we obtain the formula: BlJ (X) =[(Spec(OX [u1 t w1 , . . . , uk t wk ]int  V (u1 t w1 , . . . , uk t wk ))/Gm ] =

.

= [(Spec(OX [u1 t a1 , . . . , uk t ak | a1 ≤ w1 ])  V (u1 t w1 , . . . , uk t wk ))/Gm ]. Adjoining the generator .t −1 does not change the quotient space and leads to the formula BlJ (X) := [(Spec(OX [t −1 , J t])X )  V (J t))/Gm ]

.

= [(Spec(OX [t −1 , u1 t w1 , . . . , uk t wk ]  V (u1 t w1 , . . . , uk t wk ))/Gm ], considered in the stack-theoretic context in [36].

2.2.3 Stack-Theoretic Blow-Ups vs Cobordant Blow-Ups The two approaches: the original one with the stack-theoretic blow-ups from [27] and [4], and the one with cobordant blow-ups from [49] are to a great extent equivalent. It is a matter of personal preference which method to use. The cobordant approach carries a richer information hidden in the torus action and the space B with the vertex .Vert(B) ⊂ B playing an important role in the resolution, and used in direct computations (see Sects. 4.4.18 and 5.3, 5.4). It is also more suited for practical implementations and does not require stacks. On the other hand, the stack-theoretic blow-ups define birational transformations and do not modify smooth loci. Consequently, the stack-theoretic blow-ups use a smaller number

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of unknowns which is the dimension of the smooth ambient space. Also, the results are stated within a more general category of stacks. In the cobordant blow-ups, each operation introduces a new variable, and the smooth locus .Xsm is always multiplied by a torus, so we obtain a trivial T -bundle over .Xsm . Lastly, by applying stack theoretic quotients to weighted cobordant blow-ups by the torus action we obtain the stack-theoretic ones.

3

Cobordant Blow-Ups of Rees Algebras

3.1

Rational Rees Algebras

The algorithm in this paper will be presented in the language of the rational Rees algebras. This language is more convenient for direct computations and it efficiently simplifies the presentation of the resolution process. In particular, the resolution invariant is defined in this language. Moreover an easy computable center of the cobordant blow-up is defined by the recursive formula in the language of Rees algebras in Sect. 4.4.1, which is illustrated by Examples in Sects. 4.4.1 and 4.7. This direct explicit construction automatically implies uniqueness of the locally defined center, and enables gluing the local resolutions into global one. Arguably, the uniqueness of the construction is one of the most difficult parts in embedded resolution problems. It is usually achieved by the inductive process and comparing directly different steps of construction either by introducing relations via Hironaka’s trick or by using the homogenization-tuning method of [46] and functoriality properties. One of the advantages of Rees algebras is that their generators lie in different rational gradations. This makes the computations of the centers very straightforward. The process avoids various multiple rescalings which often lead to the increased computational complexity. The approach when generators are naturally associated with different rational weights also becomes handy when resolving some classes of singularities in positive characteristic [49]. It shall be noted that Bierstone-Milman in their resolution algorithm used rational gradations in their resolution approach [9, 10]. Their centers, however are described by the inductive process, which relies on factorization of the accumulated monomials, and as such is more technically involved. As with the other approaches utilizing Hironaka’s trick their centers are defined via the classes of a certain equivalence relation.

3.1.1 Rational Rees Algebra We shall consider here Rees algebras on a scheme X with gradations given by a finitely generated additive subsemigoups .Γ of .Q≥0 .

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Definition 3.1 By a rational Rees algebra or simply Rees algebra we mean a finitely generated .OX -algebra which can be written of the form: R=



.

Ra t a ⊂ OX [t 1/wR ],

a∈Γ

where .wR ∈ Q>0 is the smallest rational number such that .Γ = ΓR ⊆ (1/wR ) · Z, and the ideals .Ra ⊆ OX satisfy (1) .R0 = OX (2) .Ra · Rb ⊆ Ra+b If R is a rational Rees algebra, and w is a multiple of .wR then .R ext := R[t −1/w ] will be called an extended Rees algebra.  By the integral closure .R int of a rational Rees algebra .R = a∈Γ Ra t a we shall mean its integral closure in .OX [t 1/wR ]. By the vanishing locus .V (R) (or .V (R ext )) we mean V (R) = V (



.

Ra ) in Spec(R)

a>0

 (respectively .V (R ext ) = V ( a>0 Raext ) in Spec(R ext )). Remark 3.2 We do not assume here that .Ra ⊆ Rb if .a ≥ b for .a, b ∈ Γ . However this condition is satisfied if .R = R int is integrally closed in .OX [t 1/wR ].

3.1.2 Rational Rees Algebras and Q-Ideals  n n With any ideal .I one associates the .Z-graded Rees algebra .AI := OX [It] = I t , and extended Rees algebra Aext = OX [t −1 , It].

.

Similarly with a .Q-ideal .J = I 1/n we associate .AJ := OX [It n ]int ⊂ OX [t], and −1 n int −1 J = OX [t , It ] ⊂ OX [t, t ]. As we mentioned before the main idea of the rational Rees algebras is to enlarge the gradation so we obtain a simpler presentation of the graded algebra, and thus a nicer presentation of the blow-up of .I. 1/w 1/w If .J = (x1 1 , . . . , xk k ), is a .Q-ideal with all .wi ∈ N then the associated algebra w w int is of particularly simple form. However in a more general .AJ = OX [x1 t 1 , . . . , xk t k ] case the .Q-ideal .J = (x1a1 , . . . , xkak ) determines the algebra .AJ = (OX [J ])X which does not have a nice presentation. ext .A

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In such a case it is more natural to associate with .J = (x1a1 , . . . , xkak ) the rational Rees algebra A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int ⊂ OX [t 1/wA ],

.

with a rational but simpler graded presentation. Here wA := lcm(a1 , . . . , ak )

.

is the smallest positive rational such that .wA /a1 , . . . , wA /ak are all integers.

3.1.3 Rees Centers By a Rees center we mean a Rees algebra .A locally of the form A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int ,

.

for some positive rational .a1 , . . . , ak , where the integral closure is considered in OX [t 1/wA ], where .wA = lcm(a1 , . . . , ak ), and .x1 , . . . , xk is a local partial system of coordinates. Per analogy with the correspondence .J → AJ established in [4] for .J = 1/w 1/w (x1 1 , . . . , xk k ), with .wi being integral, and .ai = 1/wi , we associate with .Q-ideal ak a1 .(x , . . . , x ) the rational Rees algebra k 1 .

A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int ,

.

It is convenient to write the centers of the cobordant blow-ups in a more general extended form. By the extended center which is also called center we shall mean the extended Rees algebra Aext = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ],

.

where w is a multiple of .wA , so .w/ai are all positive integers. Again we shall often use a shorter notation A := (x 1 t 1/a1 , . . . , x k t 1/ak )

.

for the centers.

Aext := (t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak )

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3.1.4 Rescaling Definition 3.3 Given w0 ∈ Q>0 , by the t → t w0 rescaling of the rational Rees algebra  R = a∈Γ Ra t a ⊂ OX [t 1/wR ] we mean the Rees algebra R w0 =



.

Ra t w0 a ⊂ OX [t w0 /wR ],

a∈Γ

Lemma 3.4 There is a natural isomorphism R → R w0 f t a → f t w0 a . In particular, R ♣ is integrally closed in OX [t 1/wR ] iff R w is integrally closed in OX [t w0 /wR ]. Lemma 3.5 Let A := OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int and Aext := OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ],

.

where w is a multiple of wA . Then the integral closure of A in OX [t 1/w ] is equal to AInt := Aext ≥0 . Moreover if V (A) is irreducible then there is a unique monomial associated valuation νA such that νA (xi ) = 1/ai , and Aa = {f ∈ OX | νA (f ) ≥ a}.

.

Proof It is a direct consequence of Lemma 2.4, obtained by rescaling t → t 1/w .

♣

3.1.5 Cobordant Blow-Ups of Rees Centers We shall rewrite Definition 2.9 of cobordant blow-ups in the language of the Rees centers Definition 3.6 ([49]) By the full cobordant blow-up of a Rees center Aext = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]

.

we mean B = Spec(OX [t −1 , x1 t w1 , . . . , xk t wk ]),

.

where .wi = wA /ai . Here the algebra .OB is obtained by rescaling .t → t w of .Aext . By the cobordant blow-up of .Aext we mean the morphism .B+ → X, where .B+ := B  Vert(B) and .Vert(B) := V (x1 t w1 , . . . , xk t wk ) is called vertex of B. Correspondingly B− = B  VB (t −1 ) = X × Gm

.

is the trivial cobordant blow-up, where .E = VB (t −1 ) is the exceptional divisor.

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3.1.6 Exceptional Divisor In order to distinguish the dummy variable t in the center from the coordinate on B we sometimes write .tB instead of t in the formula for .B = Spec(OX [tB−1 , x1 tBw1 , . . . , xk tBwk ]). Lemma 3.7 The cobordant blow-up .B+ transforms the center 1

1

A = OX [x1 t a1 , . . . , xk t ak ]int

.

into (A · OB+ )int = OB+ [tB−1 t 1/wA ]

.

Consequently it takes the integral algebra OX [x1 t w1 , . . . , xk t wk ]int

.

into the algebra .OB+ [tB−1 t] of the ideal of the exceptional divisor .E = VB+ (tB−1 ) on .B+ . Proof We have 1

1

(A · OB+ )int = OB+ [(x1 tBw1 )tB−w1 t a1 , . . . , (xk tBw1 )tB−wk t ak ]int

.

Note however that .B+ is covered by .Bxi t wi , and on each .Bxi t wi the algebra .Aext contains 1

(t 1/wA tB−1 )wi = tB−wi t ai ∈ (A · OB+ )int . Consequently .t 1/wA tB−1 is in .(A · OB+ )int and .♣ clearly generates it.

.

4

SNC Resolution and Principalization

4.1

The Main Invariant

4.1.1 SNC-Divisors By a simple normal crossing divisor or shortly SNC divisor D on a regular scheme X we mean a Cartier effective divisor which can be locally written as D = div(ua11 · . . . · uakk ),

.

where .u1 , . . . , uk is a partial system of local parameters on X, so for any point .p ∈ V (u1 , . . . , uk ) the functions .u1 , . . . , uk ∈ OX,p determine linearly independent vectors in the cotangent space .mp /m2p .

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Remark 4.1 The SNC divisors naturally occur as the exceptional divisors of the transformations in the standard Hironaka embedded desingularization with smooth centers. To ensure the SNC condition, the smooth centers of blow-ups should have simple normal crossings with the exceptional divisor defined by the previous blow-ups. This creates additional problems and leads to a certain filtration on the logarithmic structure. The resulting invariant is consequently more technical and reflects the weights and the divisors occurring on the different steps of the algorithm. The use of weighted centers eliminates all those technicalities; the only structure needed for our algorithm is given by the SNC divisor.

4.1.2 Compatibility with SNC Divisors Definition 4.2 ([49]) Let X be a regular scheme and E be an SNC divisor. We say that a partial set of local parameters (or coordinates) x1 , . . . , xk is compatible with E at a point p (respectively on an open subset U ) if every irreducible component D of E through p (respectively on U ) is defined as V (xi ) for some coordinate xi . If V (xi ) is a component of E then we call the coordinate xi divisorial at p (resp. on U ), and write xi ∈ Ep (resp. on xi ∈ E|U ). Otherwise, we shall call xi a free coordinate at p (or on U ). We say that a Rees center A is is compatible with E at p or on U if it can be written as A = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]int or

.

Aext = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]

where x1 , . . . , xk is compatible with E at p or respectively on U . Lemma 4.3 ([49]) If Aext = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]

.

of Aext with E is compatible on smooth X , and σ : B → X is a full cobordant blow-up ♣ of Aext then σ −1 (E) is an SNC divisor on B. ai

ai

Proof The divisor E is defined locally by the product of divisorial unknowns xi1 1 ·. . .·xik k . Then its inverse image is given by (xi 1 )ai1 · . . . · (xi k )aik · t −(ai1 wi1 +...+ai1 wik ) ,

.

where xi 1 describes the strict transform of V (xi ), and t −1 determines the exceptional divisor.

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4.1.3 The Canonical Invariant We introduce the total order on the set of symbols Q+ := Q  {a+ | a ∈ Q},

.

by putting .a+ > a, and .a+ < b if .a < b. Let E be an SNC divisor on smooth X, and .I be an ideal on X. Let .Ep be the set of the divisorial coordinates at p. Let .A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int be compatible with E at a point p. We shall associate with a divisorial coordinate .xi ∈ Ep in the presentation of .A the symbol .bi := ai+ and with a free coordinate .xi simply .bi := ai . This way we will associate with the center .A the invariant inv(A) = (b1 , . . . , bk ).

.

We shall always assume that b1 ≤ . . . ≤ bk

.

by rearranging the coordinates accordingly. Remark 4.4 The main idea of using weights in .Q+ is to think of the divisorial coordinates as infinitesimally heavier and thus “more singular ” than free coordinates giving the first ones weight .1+ and the second ones weight 1. Consider the lexicographic order on ((Q+ )≥0 )≤n :=



.

((Q+ )≥0 )k .

k≤n

We compare the sequences of different lengths lexicographically by placing a se- quence of .∞ at the end. Definition 4.5 ([49]) For a point .p ∈ X we shall define the canonical invariant .invp (I) of .I at p to be invp (I) := max{(b1 , . . . , bk ) | It ⊂ OX [x1 t 1/a1 , . . . , xk t 1/ak ]int | (b1 ≤ . . . ≤ bk )}.

.

≤n if it exists. where .(x1 , . . . , xk ) are the compatible with E at p. Then .invp (I) ∈ (Q+ ≥0 ) If .invp (I) = inv(A) = (b1 , . . . , bk ) then we will say that the Rees center

A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int

.

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such that .Ip ⊆ OX [x1 t 1/a1 , . . . , xk t 1/ak ]int is a maximal admissible center at p compatible with E. The invariant .invp (I) was used in [4] in the equivalent form of .Q-ideals, and in the case where no divisors are present giving the embedded desingularization without the divisorial SNC structure. On the other hand .invp (I) can be also linked to the so called “year zero invariant” in the Hironaka resolution algorithm with smooth centers. The year zero invariant is a part of the actual invariant in the particular situation when no exceptional divisors are preset, so at the beginning of the process. The actual invariant used i is more involved and requires additional datum.

4.1.4 Presentation of Centers We shall write centers in a more compact form: OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int ,

.

where .0 < a1 < a2 . . . < ak , and .x 1 , . . . , x k is a partial system of local parameters compatible with E on open U intersecting .V (x 1 , . . . , x k ) and each .x i := (xi1 , . . . , xiki ). Then we associate with each block .x i t 1/ai the invariant .bi = (bi1 , . . . , biki ), where .bi1 ≤ . . . ≤ biki , such that .bij = ai if .xij is free on U and .bij = ai+ if .xij is divisorial on U . Consequently we can rewrite the definition of the resolution invariant in the form which is more convenient for computations and presentation. invp (I) = max{(b1 , . . . , bk ) | It ⊂ OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int }

.

We will show that the number .a1 in the maximal admissible center is simply the order of ideal .a1 = ordp (I) (Lemmas 4.15 and 4.16). We shall also use the following auxiliary invariant which generalizes the order: Definition 4.6 inv1p (R) := max{b1 | R ⊂ OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int }.

.

Similarly denote the invariants associate with the centers A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int

.

as inv(A) = (b1 , . . . , bk ),

.

inv1p (A) := (b1 )

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Remark 4.7 It is not clear a priori, that .invp (I) is well defined since the maximum may not be attained. It will be proven in Sect. 4.4.1 that there exists unique maximal admissible algebra .OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int for which the maximum .(b1 , . . . , bk ) is attained.

4.1.5 Admissibility Condition for Ideals Lemma 4.8 ([49]) The following conditions are equivalent in a neighborhood of p ∈ X for an ideal I: (1) (2) (3) (4)

It ⊂ A := OX [x1 t 1/a1 , . . . , xk t 1/ak ]int It ⊂ Aext := OX [t −1/wA , x1 t 1/a1 , . . . , xk t 1/ak ] It wA ⊂ AwA = OX [x1 t w1 , . . . , xk t wk ]int , where wi = wA /ai It wA ⊂ (Aext )wA = OX [t −1 , x1 t w1 , . . . , xk t wk ]

Proof The condition (3) and (4) are obtained by rescaling the conditions in (1) and (2). On the other hand, by Lemma 2.4, (Aext )≥0 = OX [t −1 , x1 t w1 , . . . , xk t wk ]≥0 = OX [x1 t w1 , . . . , xk t wk ]int = A.

.

This implies that condition (1) and (2) are equivalent.

♣

Definition 4.9 ([4, 49]) We say that the Rees center A = OX [x1 t w1 , . . . , xk t wk ]int is admissible for an ideal I is if It ⊂ A := OX [x1 t 1/a1 , . . . , xk t 1/ak ]int .

4.1.6 Admissibility of Rees Algebras We extend this definition to Rees algebras: Definition 4.10 ([49]) We say that the Rees center .A = OX [x1 t w1 , . . . , xk t wk ]int is admissible for a Rees algebra R or simply R-admissible if one of the equivalent condition holds: (1) .R ⊆ OX [x1 t 1/a1 , . . . , xk t 1/ak ]Int , where the inclusion is considered with the integral closure “Int” taken in the smallest ring .OX [t 1/wR,A ] containing both algebras, or, equivalently in .OX [t 1/w ] for any multiple .w = n · wR,A or for a sufficiently divisible w. (2) .R ⊂ OX [t −1/wR,A , x1 t 1/a1 , . . . , xk t 1/ak ]. (3) .R ⊂ OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ], where .w = n · wR,A . Remark 4.11 Note that the inclusion is independent of the choice of sufficiently divisible w.

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4.1.7 Resolution Invariant of Rees Algebras [49] Consequently we define the resolution invariant for Rees algebras extending Definition 4.5: invp (R) := max{(b1 , . . . , bk ) | R ⊂ OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int }

.

We show in Sect. 4.4.1 that there exists unique center .OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int for which the maximum .(b1 , . . . , bk ) is attained. We shall call it the maximal admissible center for R. Remark 4.12 In the actual resolution process of the ideals all the occurring Rees algebras R satisfy .wR,A = wA and thus the integral closure “Int” and “int” coincide.

4.1.8 Order of Rees Algebras Definition 4.13 ([49]) Let X be a regular scheme. By the order of the Rees algebra R =  a a∈Γ Ra t at the point p ∈ X, we mean ordp (R) := min {ordp (Ra )/a}.

.

a∈Γ 0

Since R is finitely generated OX -algebra the order is attained for a certain homogenous element f t p ∈ Ra t a , so that ordp (R) = ordp (f t a ) = ordp (f )/a.

.

This definition is a generalization of the order of an ideal: ordp (I) = ordp (OX [It]).

.

Immediately from the definition we see that Lemma 4.14 ordp (R w ) = w · ordp (R).

.

♣ If R = OX [It] then ordp (I) = ordp (R). Moreover we have: Lemma 4.15 ordp (I) = max{a1 ∈ Q>0 |

.

It ⊂ OX [mp t 1/a1 ]}.

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Proof We can write: 1 ordp (I) = max{a1 ∈ N | Ip ⊂ map1 } = max{a1 ∈ Q>0 | Ip ⊂ ma p }.

.

On the other hand the condition by Lemma 3.5, Ip t ⊂ OX [mp t 1/a1 ] means that Ip ⊂ a  mp 1 . ♣ One can extend this observations to the Rees algebras: Lemma 4.16 ([49]) If R ⊆ A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int then a1 ≤ ordp (R). On the other hand ordp (R) = max{a1 ∈ Q | R ⊂ OX [mp t 1/a1 ]Int }.

.

In particular if A is maximal admissible for R at p then ordp (R) = a1 . Proof By the assumption, 

R=

.

Ra t a ⊆ A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int ⊆ OX [mp t 1/a1 ]Int .

Rescaling t → t a1 gives R a1 = definition, for any a ∈ ΓR ,



aa1 

Ra t aa1 ⊆ OX [mp t]Int . Thus Ra ⊆ mp

whence, by

1  t aa1 ) ≥ 1, (1/a1 )ordp (Ra t a ) = ordp (Ra t aa1 ) ≥ ordp (maa p

.

and ordp (R) ≥ a1 . On the other hand, if ordp (R) = a1 then for any a ∈ Γ  {0} we have ordp (Ra ) ≥ aa1 , aa  so Ra ⊆ mp 1 or Ra t a ⊆ mpaa1  t a ∈ OX [mp t 1/a1 ]Int .

.

♣

4.1.9 Replacement Lemma We shall need the following result: Lemma 4.17 ([4] (Non-divisorial Case, .i = 1), [49]) Let .A = OX [x 1 t 1/a1 , . . . x k t 1/ak ]int be a center and .p ∈ V (A) be a point. Let .x 1 , . . . , x i−1 , x be a system of local parameters compatible with E at a point p, for some .i ≤ k, such that x t ai ∈ A = OX [x 1 t 1/a1 , . . . x k t 1/ak ]int

.

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in a neighborhood of p then one can find the coordinates .x 1 , . . . , x i−1 , x i , x i+1 . . . , x k such that .x ⊂ x i and A = OX [x1 t 1/a1 , . . . , x i−1 t 1/ai−1 , x i t 1/ai , x i+1 t 1/ai+1 . . . , x k t 1/ak ]int

.

in a neighborhood of .p ∈ X. Proof By Lemma 3.5, .Aai ⊂ (x 1 , . . . , x i ) + m2p . Thus upon the coordinate change of .x i compatible with E in image in .mp /(m2p + (x 1 , . . . , x i−1 ), of the set of coordinates .x is a subset of the image of coordinates in .x i . So one can extend .x, and assume that .x and 2  .x i define the same images in .mp /(mp + (x 1 , . . . , x i−1 ). Then in the completion .O X,p we can write equality of the vectors of the coordinates x i = x i + g,

.

x j = x j ,

j = i,

where the coordinates of vector .g are in .Aa1 ∩ (m2p + (x 1 , . . . , x i−1 )). This determines an    automorphism of .O X,p which takes .O X,p · A onto .O X,p · A, and determines the desired ♣

coordinate change.

.

Corollary 4.18 ([4, 49]) Assume that a center .A has two different presentations:

A = OX [x 1 t 1/a1 , . . . x k t 1/ak ]int = OX [x 1 t 1/a1 , . . . x k t 1/ak ]int

.





then the associated invariants .(b1 , . . . , bk ) = (b1 , . . . , bk ) are the same. Proof By Lemma 4.16, .a1 = a1 = ordp (A), which can be verified for generators. By Lemma 4.17, applied to both presentations, we can assume that .x1 = x1 . Restricting both .♣ algebras to .V (x 1 ) = V (x 1 ) we get the equality by the inductive assumption.

4.2

Coefficient Ideal for Rees Algebras

4.2.1 Splitting of Derivations and Compatibility In the remaining part of Sect. 4 we consider mostly varieties over a field K of characteristic 0. Definition 4.19 We say that a closed subscheme H of a scheme X splits if the embedding i : H → X splits, so there is an affine morphism .π : X → H which is a left inverse of i with .π i = idH .

.

The splitting condition implies that there is an injective morphism .OH → π∗ (OX ) of sheaves on .H.

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X,p ) splits. Example 4.20 Any regular subscheme .H ⊂ Spec(O Definition 4.21 Given a partial system of coordinates .x 1 = (x11 , . . . , x1s1 ) on a smooth X,x ), where X is smooth, we say that a system of derivations variety X or on .Spec(O .Dx 1 := (Dx11 , . . . , Dx1s ) is compatible with a Rees center .A on X if there is a certain 1 presentation A = OX [x 1 t 1/a1 , . . . , xk t 1/ak ]int ,

.

such that .Dx1i1 is computed for the complete coordinate system .x 1 , . . . , x n , extending x1, . . . , xk .

.

Definition 4.22 ([49]) Let .Dx be a system of derivations for a certain complete coordinate system containing partial system of coordinates .x on a regular scheme X. We say that .Dx splits in .OX , if .H = V (x) ⊂ X splits and the derivations in .Dx i vanish on .OH ⊂ OX . Lemma 4.23 ([49]) Let .A = OX [x 1 t 1/a1 , . . . , xk t 1/ak ]int be a center. Let .Dx 1 be a system of derivatives for a certain complete coordinate system. If .Dx 1 splits in .OX then it is compatible with the center .A so there is a certain presentation A = OX [x 1 t 1/a1 , x 2 t 1/a2 , . . . , x k t 1/ak ]int ,

.

for .x 1 , x 2 , . . . , x k such that .Dx 1 is computed for a coordinate system .x 1 , x 2 , . . . , x n extending .x 1 , x 2 , . . . , x k . Proof Consider .x i := x i|H1 . Then .x i − x i ∈ (x 1 ), and since .ai > a1 , we have that 1/ai ∈ A. So .x 1 t x i t 1/ai − x i t 1/ai ∈ (x 1 t 1/ai ) ⊂ A.

.

Hence .x i t 1/ai ∈ A and we have inclusion of the centers .OX [x 1 t 1/a1 , . . . , xk t 1/ak ]int ⊆ A. .♣ By symmetry we have the reverse inclusion. Corollary 4.24 ([49]) If X is smooth over a field K then any derivation .Dx 1 splits in  .O X,p so it is compatible with any center of the form 1/a1  , . . . , x k t 1/ak ]int . A=O X,p [x 1 t

.

♣

.

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Lemma 4.25 If .H = V (x) is a smooth subvariety that splits in a smooth variety X then there is a system of derivatives .Dx which splits in .OX . Proof We extend .x by adjoining a system of coordinates in .OH ⊂ OX .

♣

.

Remark 4.26 If .Dx splits in .OX then any function f can be written in the coefficient form  f = cα x α , where .cα ∈ OV (x) ⊂ OX so we can obtain a particularly nice description X,p . of the center on X. This can be always done in the completion .O

.

4.2.2 Derivations on the Rees Centers When considering an ideal .I of order .a ∈ N at a point .p ∈ X we can place it in a a .t -gradation of a certain .Z-graded Rees algebra .R = OX [It ] so that we have the admissibility condition R = OX [It a ] ⊂ OX [mp t],

.

with respect to maximal ideal .mp ⊂ OX,p . In such a case the coordinates are naturally associated with gradation t, and the derivatives .Dxi lower the order of ideals by 1 and are associated with gradation .t −1 . The corresponding graded derivations .t −1 Dxi act on the elements with positive gradations of .R ⊂ OX [t]. Moreover they preserve .OX [mp t]. In our approach we place the ideal .I in gradation t, and thus the corresponding coordinates and derivations shall be rescaled accordingly:  Let .R = Ra ⊂ OX [t 1/w ] be any Rees algebra on X. For any local coordinate system .x = (x1 , . . . , xn ) compatible with E on a smooth variety X, and a given .a1 ∈ Q one can introduce the graded derivations Dxi t 1/a1 := t −1/a1 Dxi

.

acting on the elements of .Ra of the gradations .a ≥ 1/a1 . More specifically if .f t a ∈ OX [t 1/w ], where .a ≥ 1/a1 then Dxi t 1/a1 (f t a ) = Dxi (f )t a−1/a1 ∈ OX [t 1/w ].

.

The idea is the following if R ⊆ A = OX [x 1 t 1/a1 , . . . , xk t 1/ak ]Int

.

then when applying the differential operators .Dx 1 t 1/a1 to the both sides we preserve the right side while enlarging the left side by adding new elements. This way we step by step enlarge R which at some point of the algorithm becomes equal to the center .A on the right side.

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More generally consider differential operators: α −|α|/ai Dxt Dx α , 1/a1 := t

.

∂ |α| 1 · α1 , α1 ! . . . αk ! ∂x1 . . . ∂xnαn

Dx α :=

.

where |α| = α1 + . . . + αn .

acting on elements .f t a ∈ OX [t 1/w ] and .a ≥ |α|/a1 : a−(|α|/a1 ) a α Dxt 1/a1 f t = Dx α (f )t

.

Example 4.27 If .f t = (x 2 + xy + y n )t, and .a1 = 2 then Dxt 1/3 ((x 3 + xy + y n )t) = (3x 2 + y)t 2/3 .

.

Lemma 4.28 Let .A = OX [x1 t 1/a1 , . . . , xk t 1/ak ]int , and .x = (x1 , . . . , xn ), where .n ≥ k, be a complete coordinate system. If .f t a ∈ Aa t a , and .|α|/a1 < a then a a−(|α|/a1 ) α , Dxt 1/a1 (f t ) ∈ Aa−(|α|/a1 ) t

.

Proof The property can be verified on monomials .x1b1 · . . . · xkbk t a ∈ Aa and on the differential operators .Dxi t 1/a1 for .i = 1, . . . , k. If .x1b1 · . . . · xkbk t a ∈ Aa then .b1 /a1 + . . . bk /ak ≥ a. So b1 /a1 + . . . + (bi − 1)/ai + . . . bk /ak ≥ a − (1/a1 )

.

and Dxi t 1/a1 (x1b1 · . . . · xkbk · t a ) ∼ (x1b1 · . . . · xibi −1 · . . . · xkbk · t a−(1/a1 ) ) ∈ Aa−(1/a1 )

.

♣

.

4.2.3 Coefficient Ideal of Rees Algebra The concept of the coefficient ideals goes back to Abhyankhar and Hironaka [23]. Since then, many different definitions have been considered in [10, 26, 43, 46] and others. In our approach the coefficient ideal is a tool to obtain a better approximation of a maximal admissible center. In fact, as it is shown in Sect. 4.4.1, the maximal admissible center .A is obtained by applying recursive coefficient ideals to a given Rees algebra .R = OX [It]. Our definition of the coefficient ideal is a priori non-canonical and easy to compute in the X,p in a so called split form (Sect. 4.2.4). This form of the coefficient ideal is completion .O much simpler and more efficient for computations and can be related to Bierstone-Milman

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who originally used a similar idea in their resolution algorithm in [9, 10] in a different setup. Definition 4.29 ([49]) Let .a1 ∈ Q>0 , and .R = OX [fj t bj ] be a Rees algebra generated by .fj t bj for .j = 1, . . . , s and let .x be any partial system of local coordinates compatible with E at p, and set .H := V (x). By the coefficient ideal with respect to .xt 1/a1 we mean the Rees algebra: bj α Cxt 1/a1 (R) := OX [( xt 1/a1 )Int , Dxt 1/a1 (fj t ),

.

|α| < bj a1 , j = 1, . . . , s ].

Here the part .OX [ xt 1/a1 ]Int , means the integral closure of .OX [ xt 1/a1 ] in the smallest subalgebra .OX [t 1/w ] containing all the generators in .Cxt 1/a1 (R). Remark 4.30 The main idea of the coefficient ideal as well as the role of the graded local parameters .x 1 t 1/a1 is given by Lemma 4.36. The coefficient ideal preserves admissibility property, while containing already some elements .x 1 t 1/a1 from a to be constructed maximal admissible center .A.

4.2.4 Coefficient Ideal in the Split Form Computing of the coefficient ideal is very straightforward when using a splitting deriva X,p or to a tions .Dx in .OX . This can be always done passing to the completion .O b corresponding étale neighborhood. For any element .fj t j ∈ Rbj , one can write .fj in X,p as .O fj =

.



cj α x α =

 |α| a1 } = V (T 1 (R)).

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Proof (1) .ordq (R) ≥ a1 if and only if .ordq (fj ) ≥ bj a1 for any j iff .Dx α (fj ) vanish at q .♣ for any integral .|α| < bj a1 . (2) The reasoning is similar.

4.3.3 Maximal Contact of Rees Algebra The concept of the hypersurfaces of maximal contact was originated by Hironaka, Abhyankhar and Giraud and developed in the papers of Bierstone-Milman and Villamayor and others. If .I is an ideal of order .a1 then a maximal contact is defined to be a local ≤a−1 a−1 (I) ⊂ DX (I). Its vanishing locus .V (u) contains the set parameter .u ∈ T a (I) = DX ≤a−1 .V (D (I)) of all the points where .ordp (I) = a. X Our definition of maximal contact is meant for Rees algebras compatible with an SNC divisor E. From our perspective a maximal contact is a partial system of coordinates .x 1 occurring in any maximal R-admissible algebra A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]int

.

at a point p, where .ordp (R) = a1 . It can be characterized by Corollary 4.60. The definition of the maximal contact is very simple if no divisors are present. In such a case it is a maximal partial set of coordinates that is contained in .T a1 (R). The notion is designed for faster and more efficient computations, eliminating superfluous steps. In particular, unlike in the standard approach it includes several maximal contact coordinates at once, which simplifies the presentation of the main invariant and greatly facilitates its computation. The actual idea of multiple maximal contact is well known to many experts. It is widely used for practical implementations. Definition 4.53 ([49]) Given a Rees algebra R on X. By a partial maximal contact of .(R, a1 ) on an open affine subset U we mean a partial system of coordinates on U compatible with E: x = (x1 , . . . , xs , xs+1 , . . . , xr ).

.

such that (1) If .i ≤ s then .xi ∈ T a1 (R)(U ) and .xi is free on U . In particular . ∂x∂ i (T a1 R) = OU . (2) If .s + 1 ≤ i ≤ r, then .V (xi ) ∈ E, and . ∂x∂ i (T a1 R) = OU . If .p ∈ V (x), and .ordp (R) = a1 , and the conditions (1) and (2) are satisfied in a neighborhood of a point p then .x is a partial maximal contact of R at p. If additionally . ∂x∂ j (T a1 R) = OX at p for .j > r, where .(x1 , . . . , xn ) is a local system of parameters extending .(x1 , . . . , xr ) then we say that .x is a maximal contact of R at p.

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We shall associate partial maximal contact with respect to .(R, a) gradation .t 1/a1 and write it in the graded form .xt 1/a1 . Lemma 4.54 ([49]) Let .x 1 t 1/a1 be a maximal contact for R at p, with .ordp (R) = a1 .   Then it is a partial maximal contact for .(R, a1 ) in a certain neighborhood U of p. Proof We consider the open neighborhood, where the conditions (1), and (2) of Definition 4.53 are satisfied. Example 4.55 Let R = OX ([(x 3 + y 4 + z5 )t, (w 2 x 4 + z7 )t 2 , (y 2 + xz2 )t 2/3 , w13 t 3/4 ],

.

where .a1 = 3 = ord0 (R), as in Example 4.50. Then .T 3 (R)t 1/3 = (x, w, v, y 2 , z2 )t 1/3 , and the maximal contact at 0 is given by 1/3 in gradation .t 1/3 . It is a partial maximal contact for .(R, 3) on X. .(x, w, v)t Lemma 4.56 If .x is a partial maximal contact of .(R, a1 ) on U , then .ordq (R) ≤ a1 for any .q ∈ U . Proof By the assumption .DX (T a1 R) = OU . This implies that for any point q in U , there is .Dx α (fj ) which is invertible at q, where .fj t bj is a generator of R and .|α| = bj a1 . Thus b .ordq (fj t j ) ≤ a1 , and .ordq (R) ≤ a1 . Lemma 4.57 ([49]) (1) A maximal contact of R at .p ∈ X exists. (2) Any partial maximal contact of R at p can be extended to a maximal contact of R at p. (3) The image of a maximal contact .x at p in .mp /m2p is determined uniquely. It is the smallest subspace compatible with .Ep containing the image of .Tpa1 (R). (4) The divisorial coordinates in the maximal contact .x at p are uniquely determined and do not depend upon the maximal contact. Proof By definition, .ordp (Ra ) = aa1 for a certain a. Thus .D aa1 −1 (Ra ) ⊂ T a1 R, and aa −1 (R )) = ord (T a1 R) = 1. Denote by .I .ordp (D 1 a p E,p the ideal generated by the divisorial coordinates in .Ep (i.e. in E passing through p). Consider the image T

.

a1

R := (T a1 R + IE,p + m2p )/(IE,p + m2p )

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of .T a1 R in .mp /(IE,p + m2p ). Its basis is determined by a partial system of local free parameters .x1 , . . . , xs ∈ T a1 (R). Then we can extend it to a partial coordinate system x := (x1 , . . . , xs , xs+1 , . . . , xr ),

.

where .xi , for .i ≥ s +1 are divisorial at p with the smallest image in .mp /m2p containing the  image of .T a1 (R). It follows that for any divisorial .xj there exists an element .v = ci xi ∈ T a1 +m2p , with .cj = 0 at p, and all .xi divisorial. Moreover, .x is exactly a maximal contact of R at p. Conversely, any maximal contact of R at p can be constructed this way. a If .x is any partial maximal contact then its image in .T 1 R can be extended to a basis a1 of .T R. This way we can extend a free part of .x to a maximal contact .x . We need to show that .x contains the whole .x. If .xj ∈ x is divisorial then by the above .Dxj (v) = 0,  ai xi (mod m2p ), where .xi are all divisorial and include .xj . where .v ∈ T a1 (R) and .v = Since the image of the maximal contact .x in .mp /m2p contains necessarily the image of v it also contains the images of all .xi with .ai = 0 including the coordinate .xj . This shows that .x ⊆ x . Finally, the divisorial coordinates are exactly those occurring in presentation of some  a .v ∈ T 1 (R), where .v = ai xi (mod m2p ), with all .xi divisorial, and thus are determined uniquely. Lemma 4.58 ([49]) Let .R ⊂ AInt = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int at a point .p ∈ X such that .a1 = ordp (R). Then upon the change of the coordinate representation of .A, .x 1 contains a maximal contact .x of R at p. Proof By Lemma 4.28 the operators .t −|α|/a1 Dx α in the definition of .T a1 (R) preserve .AInt . Thus the inclusion R ⊂ AInt = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]Int

.

implies that .T a1 (R)t 1/a1 ⊂ AInt . Let .(x1 , . . . , xr ) ⊂ T a1 (R) be the free part of the maximal contact .x. Then, by Lemma 4.17, one can assume that (x1 , . . . , xr )t 1/a1 ⊆ x 1 t 1/a1 ⊆ AInt 1/a1 ,

.

after a coordinate change. Now the image of .x 1 in .mp /m2p is the same as the image of Int in .m /m2 and contains the image of .T a1 (R). Then by Lemma 4.57, it also contains .A p p 1/a1 the image of the maximal contact .x. But .x 1 is compatible with the divisorial structure, and thus it must also contain the divisorial part of the maximal contact itself, whence .x 1 ⊇ x.

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We can associate with a partial maximal contact .x of R on U the invariant inv(x) = (1, . . . , 1, 1+ , . . . 1+ ),

.

where the 1’s correspond to the free maximal coordinates in .x, and the .1+ ’s to the coordinates in .x defining the divisors on U . Lemma 4.59 ([49]) Given a Rees algebra R of order .a1 at .p ∈ X with a maximal contact x of R at p. Then

.

(1) .inv1p (R) = a1 inv(x). (2) .R ⊂ OX [xt 1/a1 , yt 1/a2 ]Int , for some .a2 > a1 , and a coordinate system .(x, y) at p compatible with E. (3) If .x is a partial maximal contact for .(R, a1 ) on U then .inv1p (R) ≤ a1 inv(x). Proof (1) and (2) If .R ⊂ OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]Int , at p, where .a1 = ordp (R) then by Lemma 4.58, .x 1 contains a maximal contact of R after a coordinate change. Thus, by Definition 4.6, .inv1p (R) ≤ a1 inv(x). Observe that since .ordp (R) ≥ a1 we have .ordp (Cxt 1/a1 R) ≥ a1 and .ordp (Cxt 1/a1 R)|H ≥ a1 . Suppose the latter order is equal to .a1 . Then, since .Cxt 1/a1 (R)|H is generated by the coefficients as in Lemma 4.31, there is a generator f ta =



.

cα x α t a ∈ Ra t a

of R, with .cα = cα (y), for the corresponding system of local parameters .(x, y), such that ordp (cα ) = a1 a − |α| for some .α. Moreover .Dy β (cα ) = D b1 bk (cα ) is invertible, for

.

y1 ,...yk

some .β, with .|β| = ordp (cα ) = a1 a − |α|. Then .Dy β Dx α (f ) is invertible and D

.

b −1

y1 1

b

,...yk k

Dx α (f ) ∈ T a1 (R),

whence .Dy1 (T a1 (R)) is invertible, which contradicts to the condition of maximal contact at a point from Definition 4.53. This shows that .ordp (Cxt 1/a1 R)|H = a2 > a1 , which implies by Lemma 4.16 that .Cxt 1/a1 |H R ∈ OH [yt 1/a2 ]. Thus, by Lemma 4.36, R ⊂ OX [xt 1/a2 , yt 1/a2 ]Int ,

.

implying that .inv1p (R) ≥ a1 inv(x) and giving the equality.

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(3) If .x is a partial maximal contact on U then for any point .p ∈ U either .ordp (A) < a1 or .ordp (A) = a1 and by Lemma 4.58, .x extends to a maximal contact .x so we have inv1p (R) = a1 inv(x ) ≤ a1 inv(x).

.

♣

.

Corollary 4.60 ([49]) Let .x be a maximal contact of R at p. Let A = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]int

.

be a maximal R-admissible center at a point .p ∈ X. Then upon the change of the coordinate representation of .A, .x 1 = x is the maximal contact of R at p. Proof By Lemma 4.58, .x 1 contains .x after the coordinate change. On the other hand by Definition 4.6 and Lemma 4.59(1) inv1p (R) = a1 (inv(x 1 )) = a1 (inv(x)),

.

whence the equality .x 1 = x.

4.3.4 Let

♣

.

Support of the Invariant

suppinv1 (R, b1 ) := {p ∈ X | inv1p (R) = b1 }

.

The following lemma illustrates a fundamental property of maximal contact in our setting: Lemma 4.61 ([49]) Let .x be a partial maximal contact of .(R, a1 ) on U , and set .b1 := a1 inv(x). Then we have suppinv1 (R, b1 ) ⊆ V (x).

.

Proof Write .x = (x1 , . . . , xr , xr+1 , . . . , xs ), where .xi are free on U for .1 ≤ i ≤ r, and divisorial for .r + 1 ≤ i ≤ s. Let .q ∈ U  V (xi ), where .xi ∈ x is free. Then a a b −1 (f ) is invertible, for some generator .f t bj ∈ R , .xi ∈ T 1 (R) = OX and .D 1 j j j bj implying that ordq (fj t bj ) ≤ (a1 bj − 1)/bj < a1 ,

.

and hence .ordq (R) < a1 , whence .inv1q (R) < b1 .

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Let .q ∈ V (x1 , . . . , xr ) ∩ U  V (xi ), where .x1 , . . . , xr are free, and .xi with .i > r is divisorial on U and .Dxi (T a1 (R)) = OX . Thus .xi is free at q since .q ∈ V (xi ), and a .Dxi (u) is invertible for a certain local parameter .u ∈ T 1 (R). Consequently u is linearly independent at q from .x1 , . . . , xr and from the other divisorial coordinates at q. Hence there are at least .r + 1 free coordinates in a partial maximal contact .(x1 , . . . , xr , u) of R at q, and by Lemma 4.59(3) inv1q (R) ≤ a1 inv1 (x1 , . . . , xr , u) < b1 = a1 inv(x).

.

We use here the property that .(r + 1)-th component of .a1 inv1 (x1 , . . . , xr , u) is .a1 while .♣ for .b1 = a1 inv(x) it is .a1+ . Example 4.62 Let .I = (x12 + (x2 + x3 )2 ) ⊂ K[x1 , x2 , x3 ], where .x1 is free, and .x2 , x3 are divisorial. Then T 2 (It) = DX (x12 + (x2 + x3 )2 ) = (x1 , (x2 + x3 ))

.

and .(x1 , x2 , x3 ) is a maximal contact at 0, and a partial maximal contact on .X = Spec([K[x1 , x2 , x3 ]). So we have inv10 (It) = 2 · inv1 (x1 , x2 , x3 ) = (2, 2+ , 2+ )

.

and

suppinv1 (It, (2, 2+ , 2+ )) ⊆ V (x1 , x2 , x3 ) = {0}.

.

When computing the invariant .inv1p (It) away from the origin for instance at .p ∈ V (x1 , (x2 + x3 )  {0} the coordinates .x2 , x3 are free and no longer divisorial, and thus one can write the given equation as .x12 +(x2 )2 , where .x2 = x2 +x3 is free and .(x1 , x2 ) ∈ T 2 (It) is a maximal contact at p giving inv1p (It) = 2 · inv1 (x1 , x2 ) = (2, 2) < (2, 2+ , 2+ ).

.

Thus the invariant drops in a neighborhood of the origin. This example is an illustration of the idea of the heavier weights .1+ > 1 associated with divisorial coordinates. Lemma 4.63 ([49]) Let .x = (x1 , . . . , xr ) be a partial maximal contact for .(R, a1 ) on an open subset U . Then (1) .x is a partial maximal contact at all points .q ∈ V (T ≥a1 (R))∩V (x), where .ordq (R) = a1 , and .q ∈ V (x). (2) .suppinv1 (R, b1 ) = V (T ≥a1 (R)) ∩ V (x) ∩ V (Dxr+1 ,...,xk (T a1 R))) is closed on U . (3) .x is a maximal contact for the points in .suppinv1 (R, b1 ).

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Proof (1) By Lemmas 4.61, 4.52, .suppinv1 (R, b1 ) ⊂ V (T ≥a1 (R)) ∩ V (x). Moreover, by Lemma 4.56, .ordq (R) = a1 for the points .q ∈ V (T ≥a1 (R)) ∩ V (x), and, by definition, .x is a partial maximal contact at q. (2) and (3) Suppose .Dxj (T a1 (R)) = OX at .q ∈ V (T ≥a1 (R)) ∩ V (x) for some .j > r. If .xj is divisorial then .(x, xj ) is a partial maximal contact at q, and, by Lemma 4.59, .invq (R) ≤ a1 inv(x, xj ) < b 1 . If .xj is free at q then there is free local parameter .u ∈ T a1 (R) such that .Dxj (u) is invertible and .(x1 , . . . , xr , u) is a partial maximal contact at q, and .invq (R) ≤ a1 inv(x, xj ) < b1 Conversely, if .q ∈ V (T ≥a1 (R)) ∩ V (x) ∩ V (Dxr+1 ,...,xk (T a1 R))) then the condition (3) of Definition 4.53 is satisfied and .x is a maximal contact at q, and .inv1q (R) = a1 (inv(x)) by Lemma 4.59. .♣

4.3.5 Maximal Contact for Nested Ideals [49] We can directly extend the definition of the maximal contact to the nested Rees algebras. Given a coordinate system .x 1 , . . . , x n of local parameters compatible with E. Let R = OX [(x1 t 1/a1 , . . . , xk t 1/ak )Int , fj t bj ]j =1,...,s ,

.

be a Rees algebra nested at .Hk := V (x 1 , . . . , x k ), with the nested order .ordp (R|Hk ) = ak+1 . Then a nested cotangent ideal .T ak+1 (R) of R is given by a

THk+1 (R) := K

.

 |α|=bj ak+1 −1

OX Dxαk+1 ,...,x n (fj )

in .t 1/ak+1 - gradation. Its restriction defines the cotangent ideal of .R|Hk on .Hk . Consequently we define a nested maximal contact for R at .Hi given by a partial system of coordinates x k+1 = (xk+1,1 , . . . , xk+1,s , xk+1,s+1 , . . . , xk+1,r ))

.

a

on X, with free coordinates .xk+1,1 , . . . , xk+1,s ∈ THk+1 (R), and divisorial .xk+1,s+1 , . . . , K .xk+1,r such that its restriction .x k+1|Hk is a maximal contact for .R|Hk . Example 4.64 Let R be given by the presentation R = OX [(x1 t 1/3 )Int , (x12 x2 + x22 + x35 )t 1/2 ].

.

Then we can also write R in the split form R = OX [(x1 t 1/3 )Int , (x22 + x35 )t 1/2 ] = OX [(x1 t 1/3 )Int , R|H1 ],

.

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so that R is strictly nested at .H1 = V (x1 ) with nested order equal to ord0 (R|H1 ) = ord0 ((x22 + x35 )t 1/2 ) = 4,

.

and the nested cotangent ideal TH41 (R) = Dx2 ,x3 (x12 x2 + x22 + x35 ) = ((x2 + x12 ), x34 )

.

in gradation .t 1/4 . Here .bj = 1/2, .ak+1 = 4, and we consider the derivations of order bj ak+1 − 1 = (1/2 · 4) − 1 = 1.

.

Then a nested maximal contact at .H1 and 0 is .(x2 + x12 ) ∈ TH41 (R). For comparison when using the split form .R = OX [(x1 t 1/3 )Int , (x22 +x35 )t 1/2 ] we obtain a nicer cotangent ideal .TH41 = (x2 , x34 ) and a maximal contact .x2 . Choice of maximal contact does not affect the computed center. Corollary 4.65 ([49]) Let R = OX [(x1 t 1/a1 , . . . , xk t 1/ak )Int , fj t bj ]j =1,...,s

.

be a Rees algebra nested at .Hk := V (x 1 , . . . , x k ), and assume that A = OX [x 1 t 1/a1 , . . . , x n t 1/an ]int ,

.

with .n ≥ k, is a maximal R- admissible center at .p ∈ X so that R ⊂ A = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]Int

.

Then the nested order .ordp (R|Hk ) = ak+1 . Let .x be a nested maximal contact of R at p and .Hk . Then upon the change of the coordinate representation of .A, .x k+1 = x is the maximal contact of R at p. X,p . Proof Since R is nested at .Hk it can be written in the split form in .O X,p · R = O X,p [(x1 t 1/a1 , . . . , xk t 1/ak )Int , R|Hk ] O

.

Consequently .A|Hk = OH [x k+1|Hk t 1/ak+1 , . . . , x n|Hk t 1/an ]int is a maximal admissible center for .R|Hk , and we can apply Lemmas 4.16 to see that .ordp (R|Hk ) = ak . By Lemma 4.17, the free coordinates in .x form a part of .x k+1 after a coordinate change. But since .x |Hk is a maximal contact for .R|Hk , we infer, by Lemma 4.57(4), that .x k+1|Hk and .x |Hk have the same divisorial coordinates, whence the equality .x k+1 = x. .♣

Weighted Resolution of Singularities. A Rees Algebra Approach

4.4

275

Effective Algorithm

4.4.1 The Algorithm [49] Assume that the center .A is maximal admissible for a certain Rees algebra R, so that R = R1 ⊆ A := OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int .

.

For an ideal .I we consider the Rees algebra .R = OX [It]. By Lemma 4.16, .ordp (R1 ) = a1 . Let .x 1 ⊆ T a1 (R1 ) be a maximal contact at .p ∈ X, and put .H1 := V (x 1 ). Then, by Lemma 4.60, we can assume that one can change the presentation of the algebra .A so that .x 1 = x 1 is a maximal contact. This will not affect the algebra .A on the right just will change its presentation. Let .R2 := Cx 1 t 1/a1 (R1 ) be the coefficient ideal. X,p in the Then, by Lemma 4.33, .R2 is nested at .H1 := V (x 1 ), and can be written in .O split form X,p R2 = O X,p [(x 1 t 1/a1 )Int , C 1/a1 (R1 )|H1 ] X,p [(x 1 t 1/a1 )Int , R2|H1 ] = O O x1t

.

Moreover by Lemma 4.45, the very same center .A = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]int , is maximal admissible for .R2 = Cx 1 t 1/a1 (R1 ) . This leads to a simple recursive procedure. Let .Ri , for .i ≥ 2, be the algebra nested at .Hi−1 = V (x 1 , . . . , x i−1 ) such that X,p Ri = O X,p [(x 1 t 1/a1 , . . . , x i−1 t 1/ai−1 )Int , Ri|Hi−1 ] ⊆ O

.

X,p A = O X,p [x 1 t 1/a1 , . . . , x k t 1/ak ]Int , ⊆O where .A is a maximal admissible center for .Ri . By Lemma 4.65, .ordp (Ri ) = ai . Find a nested maximal contact .x i for .Ri at .Hi−1 and p. Using Lemma 4.65 we can modify the coordinates, .x i , . . . , x k in the presentation of .A and assume that the maximal contact .x i = x i occurs in the presentation of .A. Then, by Lemma 4.45, Ri+1 := Cx i t 1/ai (Ri ) ⊆ AInt = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]Int ,

.

where .A on the right is a maximal admissible center for .Ri+1 . Furthermore, by X,p as Lemma 4.40, .Ri+1 is nested at .Hi = V (x 1 , . . . , x i ), and can be written in .O X,p · Ri+1 = O X,p [(x 1 t 1/a1 , . . . , x i t 1/ai )Int , Ri+1|Hi ] = O

.

X,p [(x 1 t 1/a1 , . . . , x i t 1/ai )Int , Cx i+1 (Ri )|Hi ], =O

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We continue the procedure until we reach .Rk+1 nested at .Hk = V (x 1 , . . . , x k ), such that .Rk+1|Hk = 0. Then Rk+1 = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int = AInt ,

.

with the extended Rees algebra Aext = OX [t −1/wR,A , x 1 t 1/a1 , . . . , x k t 1/ak ],

.

where .wR,A = lcm(a1 , . . . , ar , wR ). Example 4.66 Let .I = (f ) = (x 2 + y1 y2 y3 ). Write R1 = OX [It] = OX [(x 2 + y1 y2 y3 )t]

.

The cotangent ideal is T 2 (R1 ) = DX (x 2 + y1 y2 y3 ) = (x, y1 y2 , y1 y3 , y2 y3 ) ⊂ (x) + m20 .

.

We choose maximal contact .x ∈ T 2 (R1 ) in gradation .t 1/2 . The maximal contact subvariety .H1 := V (x) and the derivative .Dx split in X. Write f t = (x 2 + y1 y2 y3 )t = (xt 1/2 )2 + y1 y2 y3 t,

.

in the coefficient form with respect to .xt 1/2 . Then, by Lemma 4.32, .R2 := Cxt 1/2 (R1 ) is strictly nested at .H1 , and can be written in the split form R2 := Cxt 1/2 (R1 ) = OX [(xt 1/2 )int , (y1 y2 y3 )t],

.

generated by .(xt 1/2 )int , and the coefficient .c0 t := y1 y2 y3 t of the generator f t. Then the nested order of .R2 at .H1 = V (x) and 0 is equal to ord0 (R2|H1 ) = ord0 (y1 y2 y3 )t = 3.

.

The nested cotangent ideal of .R2 at .H1 is TH31 (y1 y2 y3 ) = Dy21 ,y2 ,y3 (y1 y2 y3 ) = (y1 , y2 , y3 )

.

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with the nested maximal contact .y := (y1 , y2 , y3 ) in gradation .t 1/3 . Then R3 := Cyt 1/3 (R2 ) = OX [xt 1/2 , yt 1/3 ]int

.

with .R3|H2 = 0 where .H2 := V (x, y1 , y2 , y3 ) = 0. Thus .R3 is the final Rees algebra and A = R3 = OX [xt 1/2 , yt 1/3 ]int

.

is the maximal .I-admissible center at 0. Example 4.67 Let .I = x 4 + y 7 + z20 . Then .R1 = OX [(x 4 + y 7 + z20 )t] is of the order 4 at the origin 0. Its maximal contact is 3 x ∈ T 4 = DX (x 4 + y 7 + z20 ) = (x, y 4 , z13 ).

.

Moreover .H1 := V (x) and .Dx split in X and .OX respectively. Then, by Lemma 4.32, we write R2 := Cxt 1/4 (R1 ) = OX [(xt 1/4 )Int , (y 7 + z20 )t]

.

in the split form, generated by .(xt 1/4 )Int , and the coefficient .c0 t = (y 7 + z20 )t of f t = (x 4 + y 7 + z20 )t = (xt 1/4 )4 + (y 7 + z20 )t

.

The nested order at .H1 and 0 is equal to .ord0 (y 7 + z20 )t = 7. A nested maximal contact is 6 y ∈ TH71 = Dy,z (y 7 + z20 ) = (y, z14 )

.

Then .H2 := V (x, y) splits again in X and .Dy splits in .OX on .H1 . So, using Lemma 4.43, we get R3 := Cyt 1/7 (R2 ) = OX [(xt 1/4 , yt 1/7 )Int , z20 t]

.

Its nested order at .H2 , and 0 is equal to .ord0 (z20 t) = 20. The nested maximal contact is 20 19 20 1/20 , and the relevant coefficient ideal is .z ∈ T H2 = Dz (z ) in gradation .t R4 := Czt 1/20 (R3 ) = OX [xt 1/4 , yt 1/7 , zt 1/20 ]int = A.

.

4.4.2 Uniqueness In the process we do not change the algebra A = OX [x 1 t 1/a1 , . . . , x k t 1 ]int ,

.

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A. We do however change the coordinates on .A which gradually become maximal contacts for the nested Rees algebra on the left. This shows that the procedure which is independent of .A leads to a unique, a priori chosen center .A which is maximal admissible for R at p.

.

4.4.3 Existence Definition 4.68 (See Also [5, Theorem 5.3.1] (in the Language of Q-ideals)), [49]) For any Rees algebra R on a smooth variety X over K with a SNC divisor E, and for any point p ∈ X there exists a uniquely determined Rees center A = OX [x 1 t 1/a1 , . . . , x k t 1 ]int

.

which is a maximal admissible for R at p. Moreover AInt = Rk+1 is obtained by the recursive procedure R1 = R, and Ri+1 = Cx i t 1/a1 (Ri ) independent of A, where (1) (2) (3) (4)

Ri is nested at Hi−1 := V (x 1 , . . . , x i−1 ), for i − 1 = 1, . . . , k, and H0 := X. ai = ordp (Ri|Hi−1 ) x i is a maximal contact for Ri|Hi−1 . invp (R) = (a1 inv(x 1 ), . . . , ak inv(x k )).

Proof The construction does not use the Rees center A on the right side of the admissibility condition at any step, and the inductive process leads to a certain Rees center A admissible for R at p. We obtain a sequence of Rees algebras Ri nested at Hi−1 = V (x 1 , . . . , x i−1 ) such that ordp (Ri|Hi−1 ) = ai , and x i is a maximal contact for int being an R -admissible center. Ri|Hi−1 , with A = Rk+1 Now, let



A = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]int ,

.

be an admissible center for R at p, and suppose that inv(A ) ≥ inv(A ). We run the recursive procedure with A on the right side. Then since a1 ≤ a1 we conclude by Lemma 4.16, that a1 = a1 . Moreover, by Lemma 4.58(2), x 1 upon the coordinate change contains a maximal contact x 1 so that inv1 (x 1 ) ≥ inv1 (x 1 ). But since inv(A ) ≥ inv(A ) , and thus inv1 (x 1 ) ≥ inv1 (x 1 ) we get the equality x 1 = x 1 . We continue this procedure step by step and show that ai = ai and x i = x i after a possible change of the coordinate int is the maximal admissible representation of A . This will give us that A = A = Rk+1 center for I at p. ♣

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4.4.4 The Inductive Principle [4, 49] Let R be a Rees algebra of order .a1 at p and a maximal contact .x 1 , and let .H1 := V (x 1 ). Then a center .A = OX [(x 1 t 1/a1 , . . . , x k t 1/ak ]int is maximal admissible for R if and only if 1/a2 , . . . , x t 1/ak ]int is maximal admissible for .C (R) .A|H1 = OH1 [(x 2 t k |H1 . So we obtain x1 the inductive formula as in [4]: invp (R) = (inv1p (R), invp (Cx 1 (R)|H1 )),

.

4.4.5 Semicontinuity of Canonical Invariant. Local Admissibility The inductive principle above determines semicontinuity of the invariant .invp , as in [4]. We use induction on .n = dim(X) and assume that the function .p → invp (R) is upper semicontinuous for any Rees algebra .R on any smooth variety H of dimension .n − 1. Let R be any Rees algebra on a smooth X of dimension n. The condition invp (R) ≥ (b1 , . . . , bk )

.

means inv1p (R) > b1

or

.

inv1p (R) = b1

and

invp (Cx 1 |H1 (R)) ≥ (b2 , . . . , bk )).

This describes a closed subset by the semicontinuity of .inv1p , followed from Lemma 4.63 and the by the inductive assumption.

4.4.6 Duality of Rees Centers [49] The Rees centers .A have a double meaning: as admissible centers with dummy variable t and as the algebras of the full cobordant blow-ups, where .t −1 is the introduced coordinate on B. This is because both notions are closely related and are represented by identical formulas up to rescaling. To avoid confusion we shall use a variable .tB instead t for the algebras on B, where the distinction between t and .tB is needed. 4.4.7 Controlled Transforms of Ideals [4, 49] Consider an ideal .I on X. We can write .I-admissibility condition as: It ⊂ Aext = OX [t −1/wA , x 1 t 1/a1 , . . . , x k t 1/ak ]

.

Rewriting this inclusion using variable .tB and rescaling we obtain I · tBwA ⊂ OB = OX [tB−1 , x 1 tBw1 , . . . , x k tBwk ] = OX [tB−1 , x 1 , . . . , x k ],

.

where .xi := xi t wi . By Lemma 2.12, the exceptional divisor on .B+ is given by .tB−1 , which is a local parameter on B. By the above the full transform .OB · I is divisible by .tB−wA since

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tBwA · OB · I = OB · tBwA · I ⊂ OB . We define the controlled transform of the ideal .I to be

.

σ c (I) := OB · tBwA · I.

.

4.4.8 Controlled Transforms of Rees Algebras and Double Gradation [49] The relation between admissibility and the controlled transforms of ideals can be stated in terms of the double gradation. The ideal .I corresponds to the gradation .It of the Rees algebra .AI = OX [It], and we can assign to .It a double gradation .OB · I · tBwA t in wA .OB [I · t B t] which can be interpreted as the controlled transform σ c (I) := OB · tBw · I ⊂ OB

.

in gradation t.  More generally if .R = Ra t a is a Rees algebra and .A is R-admissible center so that R ⊂ Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ],

.

where w is a multiple of .wR,A then rescaling .A by .tB → tBw and putting .wi := w/ai ∈ Z≥0 we obtain the cobordant blow-up of .Aext : B = Spec([tB−1 , x 1 tBw1 , . . . , x k tBwk ] = Spec OB [tB−1 , x 1 , . . . , x k ],

.

where .xi := xi tBwi . Combining both gradations on t and .tBw we can write admissibility of Rees algebras on B as  .

Ra tBa·w t a ⊂ OB [tB−1 t −1/w , x 1 tBw1 t 1/a1 , . . . , x k tBwk t 1/ak ] ⊂

⊂ OB [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ]. Thus the Rees algebra on B: σ c (R) :=

.



(OB · Ra · tBa·w )t a

will be called the controlled transform of R.

4.4.9 Cobordant Blow-Ups and Admissibility Consequently, by the above Lemma 4.69 ([49]) If .Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ]int is admissible for R , and .σ : B = OB [tB−1 , x 1 tBw1 , . . . , x k tBwk ] is a full cobordant blow-up of .Aext then the Rees

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center on B: c ext −1/w Aext , x 1 tBw1 t 1/a1 , . . . , x k tBwk t 1/ak ]int = B := σ (A ) := OB [t

.

OB [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ] 

is admissible for .σ c (R) =

OB Ra tBwa1 a t a .

Example 4.70 Let .I = (x 4 + y 7 + z20 ) as in Example 4.67. Then .A OX [xt 1/4 , yt 1/7 , zt 1/20 ]int . Since .wA = lcm(4, 7, 20) = 140 we obtain

♣

.

=

Aext = OX [t −1/140 , xt 1/4 , yt 1/7 , zt 1/20 ].

.

The full cobordant blow-ups is defined by rescaling of .Aext : B = Spec(OX [tB−1 , xtB35 , ytB20 , ztB7 ]).

.

The controlled transform of .It is σ c (I)t = (x 4 +y 7 +z20 )t ·tB140 = (xtB35 )4 +((ytB20 )7 +(ztB7 )20 )t = ((x )4 +(yt )7 +(z )20 )t.

.

4.4.10 Derivations on Cobordant Blow-Up [4, 49] Let B = Spec(OX [tB−1 , x1 tBw1 , . . . , xk tBwk ]) → X

.

be the full cobordant blow-up of a center .Aext = OX [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]. The sheaf .DX of the derivations on X is a coherent .OX -module locally generated by the derivations .Dxi . One can write derivations .Dxi on B as .Dxi = tB−wi Dxi , and .Dxj = Dxj . This observation can be interpreted using a double gradation principle: We assign with the derivations .t −1/ai Dxi in the graded form the double gradation with respect to t and .tBw to obtain the controlled transform σ c (t −1/ai Dxi ) = t −1/ai tB−wi Dxi = t −1/ai Dxj .

.

Similarly define the controlled transform of .t −1/a1 DX to be the subsheaf σ c (t −1/a1 DX ) := t −1/a1 tB−w1 OB · DX ⊆ t −1/a1 DB

.

of the sheaf .DB of the derivations on B in gradation .t −1/a1 . Consequently c −b/a1 D ≤b ) := t −b/a1 t −bw1 O · D ⊆ t −1/a1 D ≤b , for any .b ∈ Z . .σ (t B X >0 X B B

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Example 4.71 In Example 4.83 .x = xtB35 , .y = ytB20 , .z = ztB7 . So using double gradation we can write ∂ ∂ ∂ = t −1/4 ) = t −1/4 tB−35 ∂x ∂x ∂x ∂ ∂ ∂ σ c (t −1/7 ) = t −1/7 tB−20 = t −1/7 ∂y ∂y ∂y σ c (t −1/4

.

σ c (t −1/20

∂ ∂ ∂ = t −1/20 . ) = t −1/20 tB−7 ∂z ∂z ∂z

Thus σ c (t −1/4 (OX · (

.

= t −1/4 (OB (

∂ ∂ ∂ ∂ ∂ ∂ , , )) = t −1/4 tB−35 (OB ( , , )) = ∂x ∂y ∂z ∂x ∂y ∂z

∂ ∂ ∂ ∂ −15 ∂ −28 ∂ ∂ , tB , tB )) ⊂ t −1/4 (OB ( , , , −1 )). ∂y ∂z ∂x ∂y ∂z ∂t ∂x

4.4.11 The Order of the Controlled Transforms Lemma 4.72 ([4, 49]) Let σ : B → X be a cobordant blow-up of R-admissible center Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ], where ordp (R) ≤ a1 for p ∈ X. Then ordp (σ c (R)) ≤ a1 for p ∈ B. Proof Write R = OX [fj t bj ]. If ordp (R) = a1 then ordp (fj t bj ) = ordp (fj )/bj = a1 for some j , so ordp (fj ) = bj a1 . Thus there exists α, with |α| = bj a1 , such that Dx α fj is invertible. Consequently −bj w −bj

Dx α fj = (tB

.

≤bj a1

⊆ DB

t

b w

≤bj a1

Dx α )(fj tBj t bj ) ∈ σ c (t − bj DX

)(σ c (fj t bj ))

(σ c (Rbj )) = OB ,

which shows that ordp (σ c (R)) ≤ a1 .

♣

Example 4.73 In Example 4.83 the controlled transform σ c (It) = σ c (x 4 + y 7 + z20 )t = ((x )4 + (yt )7 + (z )20 )t

.

has maximal order 4.

4.4.12 Controlled Transforms of Cotangent Ideal Similarly we have Lemma 4.74 ([4, 49]) Let .σ : B → X be a cobordant blow-up of .R = OX [fj t bj ]admissible center .Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ]. If .T a1 (R) is the cotangent

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ideal for R then σ c (T a1 (R)t 1/a1 ) ⊆ (T a1 (σ c (R))t 1/a1 .

.

Proof Let .fj t bj ∈ Rbj , and .|α| = bj a1 − 1. Then .ordp (fj t bj ) = ordp (fj )/bj ≥ a1 . So .ordp (fj ) ≥ bj a1 and bj α (Dx α fj )t 1/a1 tBw1 = σ c ((Dx α fj )t 1/a1 ) = σ c (Dxt 1/a1 (fj t )) =

.

c bj a1 c α = σ c (Dxt (σ (R)t 1/a1 ). 1/a1 )(σ (fj t )) ∈ T

♣

.

4.4.13 Controlled Transforms of Partial Maximal Contact Lemma 4.75 ([4, 49]) Let σ : B → X be a cobordant blow-up of R = OX [fj t bj ]admissible center Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ]. Assume that x 1 be a partial maximal contact for (R, a1 ) on an open affine subset U then σ c (x 1 ) is a partial maximal contact for σ c (R) on σ −1 (U ). Proof σ c (x 1 ) = (x1 , . . . , xr ) is a local system of coordinates. So if xi ∈ T a1 (R) ∩ x 1 is free then xi = σ c (xi ) ∈ T a1 (σ c (R) ∩ σ c (x 1 ) is free. Thus condition (1) of Lemma 4.53 is satisfied. If xi ∈ x 1 is divisorial, then using Lemma 4.74, we have OB = OB · Dxi T a1 (R) = σ c (Dxi t 1/a1 (T a1 (R)t 1/a1 ) =

.

t −1/a1 Dxi σ c ((T a1 (R)t 1/a1 ) ⊂ Dxi (T a1 (σ c (R)) and thus condition (2) of Lemma 4.53 is also satisfied.

♣

Example 4.76 In Example 4.83 3 T 4 (I) = T 4 (OX [It]) = DX (I) = (x, y 4 , z17 )

.

the controlled transform σ c (T 4 (I)t 1/4 ) = (x, y 4 , z17 )tB35 t 1/4 =

.

= (x , (y )4 tB−55 , (z )tB−84 )t 1/4 ⊂ (x , (y )4 , (z )17 )t 1/4 = T 4 (σ c (I))t 1/4 . Then xt 1/4 is a maximal contact for the Rees algebra R = OX [It], briefly for It at 0, and a partial maximal contact on X for (It,4). The controlled transform of xt 1/4 is σ c (xt 1/4 ) = xt 1/4 tB35 = x t 1/4

.

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is the maximal contact x t 1/4 ∈ T 4 (σ c (I))t 1/4 = (x , (y )4 , (z )17 )t 1/4

.

at V (x , y , z ) which is a partial maximal contact on B for (σ c (It), 4).

4.4.14 Controlled Transform of the Coefficient Ideal Lemma 4.77 ([49] (See Also [4])) Let σ : B → X be a cobordant blow-up of R = OX [fj t bj ]-admissible algebra A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ] then σ c (R) = OB [σ c (fj t bj )], and we have commutativity: σ c (Cx 1 t 1/a1 (R, a1 )) = Cx t 1/a1 (σ c (R), a1 ).

.

1

Proof The assertion follows from the of the commutativity of the controlled transforms with derivations: σ c (Dxα

.

1t

1/a1

(fj t bj )) = σ c (Dxα

1t

1/a1

)(σ c (fj t bj )) = Dxα t 1/a1 (σ c (fj t bj )) 1

in t |a| -gradation for |α| < bj a1 .

♣

Example 4.78 In Examples 4.67, 4.83 the coefficient ideal for R = OX [(x 4 + y 7 + z20 )t] at xt 1/4 is equal to Cxt 1/4 (R) = OX [(xt 1/4 )Int , (y 7 + z20 )t]

.

and its controlled transform is equal to σ c (Cxt 1/4 (R)) = OB [(xtB35 t 1/4 )Int , (y 7 + z20 )tB140 t] = OB [(x t 1/4 )Int , ((y )7 + (z )20 )t] =

.

=Cx t 1/4 (((x )4 + (y )7 + (z )20 )t) = Cx t 1/4 (σ c (R)).

4.4.15 Restriction of Cobordant Blow-Up to a Maximal Contact Lemma 4.79 ([4, 49]) If Aext is admissible for R then Aext |H is admissible for R|H and σ c (R)|H = σHc (R|H ), where H := V (x 1 ). The restriction of the blow-up σX : B → X of Aext to the strict transform HB = V (x 1 ) ♣ of H = V (x 1 ) is the cobordant blow-up σH : HB → H of the restriction Aext |H . 4.4.16 The Centers with Maximal Invariant Lemma 4.80 ([49] (See Also [4])) Let R be a Rees algebra, and A be a maximal admissible center for R at a point p ∈ X. Then there exists an open neighborhood U

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of p such that .

max inv(R) = invp (R) = inv(A) U

on U is attained at V (A). Proof Let x 1 be a maximal contact at p for R. Then, by Lemma 4.54, it is a partial maximal contact on an open subset U of p. Thus by Lemma 4.59(3), inv1 (R) ≤ b1 := a1 inv(x 1 ).

.

In addition, by Lemma 4.63, the maximum max inv1 (R) = b1 of the invariant inv1p (R), p ∈ X, is attained at the closed subset suppinv1 (R, b1 ) ⊂ H1 = V (x 1 ), and x 1 is a maximal contact along suppinv1 (R, b1 ). Furthermore for the points in suppinv1 (R, b1 ), we have invp (R) = (inv1p (R), invp (Cx 1 t 1/a1 (R)|H1 ),

.

Moreover A|H1 is maximal admissible for R at p. It suffices to use induction on dimension to find an open neighborhood U of p such that invp (Cx 1 t 1/a1 (R)|H1 ) attains ♣ its maximal on V (A|H1 ) = V (A).  Definition 4.81 ([4, 49]) Let R = Ra be a Rees algebra on a smooth variety X over a field K such that Ra = OX for any a ∈ A. There exists a unique center A(R) = A such that (1) The maximum max inv(R) = (b1 , . . . , bk ) of the invariant invp (R), p ∈ X, is attained at V (A). (2) A = OX [x 1 t 1/a1 , . . . , x k t 1/ak ]Int is a maximal admissible center for R, with inv(A) = (b1 , . . . , bk ), where bi = ai inv(x i ). Proof Lemma 4.80 implies that invp (R) is upper semicontinuous, and admits finitely many values. Consider the closed point set S := suppinv(R, (b1 , . . . , bk )) := {p ∈ X | inv(R) = (b1 , . . . , bk )},

.

where invp (R), where p ∈ X, attains its maximal value max inv(R) = (b1 , . . . , bk ). Then for any point p ∈ S let A be a maximal admissible center for R at p. Then V (A) = suppinv(R, (b1 , . . . , bk ))

.

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locally around p. Moreover, A is a unique maximal admissible center for R at all points of V (A) in a neighborhood of p. Thus A glues to a unique maximal admissible center for ♣ R along V (A) = suppinv(R, (b1 , . . . , bk )) as desired.

4.4.17 Cobordant Blow-Ups of the Centers with Maximal Invariant  Definition 4.82 ([49] (See Also [4])) Let R = Ra t a be a Rees algebra on a smooth variety X over a field K such that Ra = OX for any a ∈ A. Let A = A(R) be maximal admissible center for R at V (A), and B → X be a full cobordant blow-up of Aext . Then in a neighborhood of V (A) such that invp (R) attains its maximum max inv(σ c (R)) = (b1 , . . . , bk ) at V (A) we have (1) The maximum max inv(σ c (R)) = (b1 , . . . , bk ) is attained at V (σ c (A)) with the invariant inv(σ c (R)), and such that σ c (Aext ) = OB [t −1/w , x1 t 1/a1 , . . . , xk t 1/ak ]

.

is maximal admissible center for σ c (R) (2) max inv(σ c (R)) < (b1 , . . . , bk ) on B+ = B  V (σ c (A)). Proof We shall use induction on dimension of X. If dim(X) = 0 then A = R = OX = OX [0], p = X, and invp (R) = () has no entries and thus corresponds to the infinite sequence of ∞. Let p ∈ V (A), and set ordp (A) = a1 . Consider a neighborhood U of p such that x 1 is a maximal contact on U at p, and a partial maximal contact on U . Then, by Lemma 4.63, inv1 attains its maximum b1 on suppinv1 (R, b1 ) ⊂ V (x 1 ). Moreover x 1 is a maximal contact along suppinv1 (R, b1 ). Consequently for the points p ∈ suppinv1 (R, b1 ) we have invp (R) = (inv1p (R), invp (Cx 1 t 1/a1 (R)|H1 )).

.

By Lemma 4.75, the controlled transform σ c (x 1 ) = x 1 is a partial maximal contact on BU = σ −1 (U ) for σ c (R). Then, by Lemma 4.63, max inv1 (σ c (R)) ≤ b1 , and it attains its value b1 on suppinv1 (σ c (R), b1 ) ⊆ H1 = V (x 1 ),

.

and for the points p in suppinv1 (R, b1 ) we have that x 1 is a maximal contact. Moreover, by Lemma 4.77, and Sect. 4.4.4 we get invp (σ c (R)) = (inv1p (σ c (R)), invp (Cx t 1/a1 (σ c (R))|H1 ).

.

1

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By Lemma 4.79, the restriction of B → X to H1 = VB (x 1 ) is the cobordant blow-up σH1 : H1 → H1 of Aext |H1 . By the inductive assumption on dimension the conditions (1), (2) of the Proposition are satisfied on H1 for Cx 1 t 1/a1 (R)|H1 and for the cobordant blowup σH1 : H1 → H1 . Moreover, by Lemma 4.36, and the assumption, Aext |H1 is maximal admissible for Cx 1 t 1/a1 (R)|H1 on H1 , associated with max inv(Cx 1 t 1/a1 (R)|H1 ). is maximal admissible Consequently, by the inductive argument σ c (Aext )|H1 = Aext B|H1 for σHc (Cx 1 t 1/a1 (R)|H1 ) = Cx t 1/a1 (σ c (R))|H1 .

.

1

1

By Lemma 4.36 and the fact that x 1 is a partial maximal contact associated with c max inv1 (σ c (R)) = b1 we conclude that Aext B is maximal admissible for σ (R). By the inductive assumption the maximum .

max inv(σ c (Cx 1 t 1/a1 (R)|H1 )) = (b2 , . . . , bk )

on B is attained at V (σHc 1 (A|H1 )). Thus .

max inv(σ c (R)|H1 ) = max(b1 , inv(Cx 1 t 1/a1 (R)|H1 ) = (b1 , . . . , bk )

is attained at V (σHc 1 (A|H1 )) = V (σ c (A)|H1 ) = V (σ c (A)) ⊆ H1 ,

.

and σ c (A) is a maximal admissible center for σ c (R) associated with max invσ c (R).

♣

4.4.18 Resolution Principle [49] (See Also [4]) Summarizing the above, the resolution process consists of the following: Given a rational Rees algebra on a smooth X with SNC divisor E over a field K of characteristic zero. (1) The invariant .invp (R) on X is semicontinuous. (Sect. 4.4.5) (2) It attains its maximum .maxinvX (R) at a certain unique center .Aext (R) on X. (Proposition 4.81) (3) The invariant .invp (σ c (R)) on the full cobordant blow-up .B → X at .Aext (R), attains its maximum maxinvX (R) = maxinvB (σ c (R))

.

exactly at the vertex .Vert(B) = V (σ c (Aext )) of the full cobordant blow-up .B → X associated with the Rees center .σ c (Aext (R)) on B. (Proposition 4.82(1))

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(4) The invariant .invp (.) drops on the cobordant blow-up B+ = B  V (σ c (Aext (R))) = B  Vert(B)

.

after removing the vertex .Vert(B), so that maxinvX (R) > maxinvB+ (σ c (R)).

.

(Proposition 4.82(2)) Example 4.83 Let .I = (x 4 +y 7 +z20 ) as in Example 4.67, with maximal invariant attained at the origin inv0 (I · t) = (4, 7, 20)

.

with maximal admissible algebra .A = OX [xt 1/4 , yt 1/7 , zt 1/20 ]int . Then after the full cobordant blow-up .B = Spec(OX [tB−1 , xtB35 , ytB20 , ztB7 ]) at .Aext the controlled transform is σ c (I)t = ((x )4 + (y )7 + (z )20 )t

.

with maximal invariant .

max invB (σ c (I)t) = (4, 7, 20)

attained at the vertex .Vert(B) = V (x , y , z ). Thus the invariant drops on .B+ = B  Vert(B).

4.4.19 Strict Transforms Let B be the full cobordant blow-up of .Aext , and let R be an .Aext admissible cobordant blow-up. Recall that B− = B  V (t −1 ) = X × Gm → X

.

is trivial over X. By the strict transform .σ s (R) of a Rees algebra R on X under a full cobordant blow-up .σ : B → X of .A on X we mean the schematic closure of .(OB · RX )|B− = OB− · RX . Then σ s (R) := {f ∈ OB [t 1/wR ] |

.

tB−a f ∈ OB · R,

a ≥ 0}.

This implies that .σ c (R) ⊆ σ s (R), and thus .invp (σ c (R)) ≥ invp (σ s (R))) for any .p ∈ B.

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If .I is an ideal on X admissible for .A so that It ⊂ Aext = OX [t −1/w , x 1 t 1/a1 , . . . , x k t 1/ak ],

.

then its controlled transform is equal to .σ c (I) = OB tBw · I, while the strict transform is given by σ s (I) := {f ∈ OB |

.

4.5

tB−a f ∈ OB · I,

a ≥ 0}.

Properties of the Invariant

4.5.1 The Invariant at the Smooth Points [49] Assume that Y is smooth and of codimension k and is described at .p ∈ Y by the partial set of free local parameters .Y = V (u1 , . . . , uk ) compatible with E, then .OX [(u1 , . . . , uk )t] is maximal .IY - admissible center at p, with .invp (I) = (1, . . . , 1). Conversely, if .invp (I) = (1, . . . , 1) then there exists a partial system of free local parameters .u1 , . . . , uk ∈ I such that OX [IY t] ⊂ OX [(u1 , . . . , uk )t]int = OX [(u1 , . . . , uk )t].

.

So .IY = (u1 , . . . , uk ) is smooth generated by free coordinates and compatible with E.

4.5.2 Torus Action [49] If X admits a torus action and R is T -stable, then one can choose inductively semiinvariant coordinates of the maximal admissible centers. These are the nested maximal contacts in the T -stable cotangent ideals. Moreover, if additionally X admits a geometric quotient .X/T , then the number of semiinvariant coordinates in a T -stable center at a point cannot exceed .dim(X/T ) = dim(X) − dim(T ), as it describes a smooth T -stable subvariety of X of dimension at least .dim(T ). Consequently the set of values of .invp (I) in the resolution process X0 ← X1 ← X2 ← . . .

.

is contained in .(Q+ )k≥0 , where k = dim(X0 ) = . . . = dim(Xi ) − dim(Gim ).

.

4.5.3 The dcc Condition [4, 49] i Let R be a Rees algebra on X generated by the gradations .Ra i t a , for .i = 1, . . . , s. Denote by .a = lcm(a 1 , . . . , a s )). Then we see immediately that .A is R-admissible if and only if

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Ra t a ⊂ A. Thus we have

.

invp (R) = 1/a · (invp (Ra )),

.

and the problem reduces to the ideals on X. Now for any ideal .I on X we show, by the induction on .dim(X) that the set of all values .invp (I) satisfies dcc. By the inductive formula we have invp (I) = (inv1p (I), invp (σHc 1 (Cx 1 t 1/a1 (OX [(It)]|H1 )),

.

where the restricted coefficient ideal C := Cx 1 t 1/a1 OX [(It)]|H1 ) =

.



Ca/a1 · t a/a1

is generated at gradations .C1/a1 t 1/a1 , . . . , C(a1 −1)/a1 t (a1 −1)/a1 . By the argument above and induction invp (C) = 1/a1 ! · invp (C(a1 −1)! )

.

satisfies dcc. Now since .inv1p (I) satisfies dcc condition we conclude that .invp (R) satisfies dcc. Consequently by the previous section the set of values of .invp (R) in a resolution sequence also satisfies dcc.

4.5.4 Functoriality of the Invariant [4, 49] Lemma 4.84 The invariant invp (R) and the maximal admissible center A is functorial for smooth morphisms, field extensions, and group actions. Functoriality of invp (I) and the maximal admissible centers A at p is a consequence of i (I), Rees algebras, coefficient ideals, and the maximal contacts. the functoriality of DX

4.6

Final Conclusions

4.6.1 Functorial Principalization [4] (in a Non-SNC-setting) [49] Let .I be an ideal on X. We run the SNC resolution algorithm using cobordant blow-ups at the extended centers Aext = OX [t −1/wA , x 1 t 1/a1 , . . . , x 1 t 1/ak ],

.

associated with the maximal .I-admissible centers .A(I) on X. We use the controlled transforms of ideals .σ c (I). Each such a blow-up lowers .maxinvX (I). The process is

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291

continued until the maximum value of .maxinv(σ c (I)) of the controlled transform of .I drops to zero. In such a case, the controlled transform .σ c (I) of .I becomes .OX , and the full transform .OX · I is locally the product of divisorial components. Each consecutive cobordant blowup .Xi+1 → Xi will create inductively a scheme .Xi+1 with action of .Ti+1 = Ti × Gm . Since all the constructions are canonical and functorial for smooth morphisms, the centers and subscheme .Yi ⊂ Xi are automatically .Ti -stable. This proves Theorem 1.2.

4.6.2 Embedded Desingularization [4] (in a Non-SNC-setting) [49] In the process of the embedded resolution of .Y ⊂ X, we consider the ideal .I = IY of Y on X, and the Rees algebra .RY = OX [IY t] and run the algorithm applying the cobordant blow-ups at maximal .I-admissible centers .A = A(I). In the process, we use the strict transforms of the ideal .I = IY instead of the controlled transform. By Proposition 4.82 each cobordant blow-up lowers .maxinvI (X). Moreover, the set of .maxinvI (X) satisfies dcc, and thus the algorithm is finite. We run the algorithm until we reach .maxinvX (I) to be equal to .invp (I) = (1, . . . , 1). In such a case, the strict transform of Y is smooth and have SNC with the exceptional divisor which is also SNC. One can use the controlled transforms of ideals instead of the strict transforms with no change in the strategy. This proves the existence of the functorial embedded desingularization by the smooth cobordant blow-ups as in Theorem 1.3. As a result, we obtain a smooth subvariety .Y having SNC with an SNC exceptional divisor .E on .X , and with the torus action. Each consecutive cobordant blow-up .Xi+1 → Xi will create inductively a variety .Xi+1 with action of .Ti+1 = Ti × Gm . Since all the constructions are canonical and functorial for smooth morphisms, the centers, and subvariety .Yi ⊂ Xi are automatically .Ti -stable. On the level of the geometric quotients, we obtain the geometric quotients of cobordant blowups, which are weighted blow-ups. When considering stack-theoretic quotients .[Xi /Ti ] we obtain smooth stacks and the procedure described in [4]. 4.6.3 Nonembedded SNC Resolution [4] (in a Non-SNC-setting) [49] The nonembedded resolution is obtained from the embedded resolution using local embeddings and the functoriality properties. We associate with a variety Y over K initially without any divisors a modified invariant

p (Y ) in the following way. Embed Y locally into a smooth variety X. Any two such .inv embeddings .Y ⊂ X1 and .Y ⊂ X2 into smooth varieties of the same dimension are étale equivalent. If .dim(X1 ) + m = dim(X2 ), and .m ≥ 0 then the induced embeddings m and .Y ⊂ X are étale equivalent. Here the embedding .X ⊆ Am = .Y ⊆ X1 ⊆ A 2 1 X1 X1 Spec(OX [x1 , . . . , xm ]) is defined by .V (x1 , . . . , xm ). For a closed embedding .Y ⊂ X into a smooth X of dimension n, and .p ∈ Y let

p (Y ) on Y to be the equivalence class .invp (IY ) = (b1 , . . . , bk ). We define the invariant .inv of the sequences .(b1 , . . . , bk )n , marked by .n = dim(X) where .invp (IY ) = (b1 , . . . , bk ).

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We consider the equivalence relation generated by (b1 , . . . , bk )n (1, . . . , 1, b1 , . . . , bk )n+m ,

.

where m is the number of ones in front of .a1 . One can compare two equivalence classes lexicographically, as before, by fixing their representative with the same marking n. By the construction and the above, this definition is independent of the embeddings. Indeed, for any local embeddings .Y ⊂ X1 = V (x1 , . . . , xm ) ⊂ Am X1 and .Y ⊂ X2 with .dim(X1 ) + m = dim(X2 ) we get that .(x1 , . . . , xm ) is partial maximal contact in gradation X,p the maximal admissibility condition t. Then passing to the completion .O X,p · RY = O X,p [(x1 , . . . , xm )t, RY |X1 ] ⊂ O X,p · A := O X,p [x 1 t, . . . x k t 1/ak ], O

.

with .(x1 , . . . , xm ) ⊂ x 1 is equivalent to the maximal admissibility .RY |X1 ⊂ AY |X1 , and thus invp (RY,AmX ) = (1, . . . , 1, invp (RY |X1 ) =

.

1

.

= (1, . . . , 1, invp (IY,X1 ).

Moreover, by functoriality, we have invp (IY,X2 ) = invp (IY,AmX ) = (1, . . . , 1, invp (IY,X1 )),

.

1

p (IY,X2 ) is independent of the embedding.

p (IY,X1 ) = inv whence .inv

Thus we can run the embedded algorithm using the centers associated with .max inv on Y . It locally corresponds to the center .Aext associated with .maxinv(IY ) for a certain embedding .Y ⊂ X. We consider the cobordant blow-up .σ : X → X of .Aext and their

drops restriction to the strict transform .σ|Y : Y → Y . Then by Proposition 4.82, .max inv to the minimal value .(1, 1, . . . , 1)n where n is dimension of the ambient variety .X , and the number of 1-s is equal to the codimension of the smooth subvariety .Y . The exceptional SNC divisor on a local ambient space .X is compatible with a smooth subvariety .Y . Consequently, its restriction to .Y defines an SNC-exceptional divisor and determines a functorial nonembedded SNC resolution of Y . This completes the proof of Theorem 1.4.

4.7

Computations of the Centers

Example 4.85 We will illustrate the algorithm in Sect. 4.4.1 for the following example. Let I be generated by 2 8 f = u31 + u42 u73 + u11 3 u4 + u2

.

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on X = Spec(K[u1 , u2 , u3 , u4 ]). The center is obtained by the recursive formula R1 = OX [It],

.

Ri+1 = Cx i t 1/a1 (Ri ),

Aint = Rk+1

where Rk+1|Hk = 0 as in Proposition 4.68. Set R1 := OX [f t]. We check that the maximal order is 3. A maximal contact for R1 at the point p := 0 is given by 2 8 2 u1 ∈ T 3 (R1 ) = D2 (u31 + u42 u73 + u11 3 u4 + u2 ) ⊂ (u1 + mp ),

.

in the gradation t 1/3 . The derivative Du1 splits in OX and, by Lemma 4.32, R2 := Cu1 t 1/3 (R1 ) is strictly nested at H1 := V (u1 ). The generator f t can be written in the coefficient form with respect to u1 t 1/3 as 2 8 f t = (u1 t 1/3 )3 + (u42 u73 + u11 3 u4 + u2 )t

.

Then by Lemma 4.32, we write the coefficient ideal in the split form 2 8 R2 := Cu1 t 1/3 (OX [f t]) = OX [(u1 t 1/3 )Int , u42 u73 + u11 3 u4 + u2 )t]

.

8 2 1/3 )Int . generated by the unique coefficient c0 (f t) = u42 u73 + u11 3 u4 + u2 , and (u1 t The nested order of R2 at V (u1 ) and p is equal to 2 8 ordp (R2|V (u1 ) ) = ordp (f|V (u1 ) ) = ordp (u42 u73 + u11 3 u4 + u2 ) = 8.

.

A maximal contact for R2|V (u1 ) is given by 7 2 8 2 u2 ∈ TH81 (R2 ) = D(u (u42 u73 + u11 3 u4 + u2 ) ⊂ u2 + mp 2 ,...,u4 )

.

in gradation t 1/8 . Again H2 = V (u1 , u2 ) splits in X with Du1 split on H1 and we write the nested generator on H1 in the coefficient form with respect to u2 t 1/8 we obtain 2 8 1/8 8 2 (u42 u73 + u11 ) + (u2 t 1/8 )4 (u73 t 1/2 ) + u11 3 u4 + u2 )t = (u2 t 3 u4 t,

.

2 giving the coefficients cu4 · t 1/2 = u73 t 1/2 , and c0 · t = u11 3 u4 t. By Lemma 4.43, the nested 2 coefficient ideal in the split form is given by the formula: 2 8 R3 := Cu2 t 1/8 (R2 ) = Cu2 t 1/8 (OX [(u1 t 1/3 )Int , u42 u73 + u11 3 u4 + u2 )t]) =

.

2 = (OX [(u1 t 1/3 , u2 t 1/8 )Int , u73 t 1/2 , u11 3 u4 t],

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and is generated by the coefficients cu4 t 1/2 , c0 t and (u1 t 1/3 , u2 t 1/8 )Int . The nested order 2

of R3 at p ∈ H2 = V (u1 , u2 ) is equal to 13 < nested maximal contact is

7 1/2

2 = 14 and is attained for u11 3 u4 t. The

2 (u3 , u4 ) = TH132 (R3 ) = Du123 ,u4 (u11 3 u4 )

.

in gradation t 1/13 . Finally 2 R4 = C(u3 ,u4 )t 1/13 (R3 ) = C(u3 ,u4 )t 1/13 (OX [(u1 t 1/3 , u2 t 1/8 )Int , u73 t 1/2 , u11 3 u4 t]) =

.

= OX [u1 t 1/3 , u2 t 1/8 , (u3 , u4 )t 1/13 ]int , which gives the center A = R4int = (u1 t 1/3 , u2 t 1/8 , (u3 , u4 )t 1/13 ), whence invp (f ) = (3, 8, 13, 13).

.

The corresponding extended center: Aext = (t −1/312 , u1 t 1/3 , u2 t 1/8 , (u3 , u4 )t 1/13 )

.

is obtained by adjoining t −1/wA , where wA = lcm(3, 8, 13) = 312. Rescaling by t → t 312 gives B := Spec(OX [t −1 , u1 t 104 , u2 t 39 , u3 t 24 , u4 t 24 ]) =

.

Spec(K[t −1 , u1 t 104 , u2 t 39 , u3 t 24 , u4 t 24 ]), with the new coordinates t −1 , u 1 = u1 t 104 , u 2 = u2 t 39 , u 3 = u3 t 24 , u 4 = u4 t 24 . Computing the controlled transform σ c (f ) = t wA f = t 312 f

.

of f in OB in the new coordinates so that u1 = u 1 t −104 , u2 = u 2 t −39 , u3 = u 3 t −24 , u 4 = u4 t −24

.

gives σ c (f ) = ((u 1 )3 + t −12 (u 2 )4 (u 3 )7 + (u 3 )11 u24 + (u 2 )8 )

.

with the same invariant equal to (3, 8, 13, 13), and the same quasihomogeneous initial form (see Sect. 5.2.1) at the points of the vertex Vert(B) = V (u 1 , u 2 , u 3 , u 4 ). On B+ =

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B  V (u 1 , u 2 , u 3 , u 4 ) = B  Vert(B) after the vertex is removed, the maximum of the invariant invp (I) drops (see Sect. 4.4.18). We will verify this directly. We have four affine charts Bu i = B  V (u i ) on B+ , where we can easily find the equations. In particular on Bu 1 and Bu 2 the controlled transform f = σ c (f ) = t 312 f is invertible. On Bu 3 the function f = (u 1 )3 + (ξ3 )7 t −12 (u 2 )4 + (ξ3 )11 u24 + (u 2 )8 ),

.

is of order 2 where ξ3 := u 3 is a unit. On Bu 4 the maximal invariant maxinv drops to (3, 8, 11) < (3, 8, 13, 13). Example 4.86 Let f = x α1 1 + . . . + x αnn

.

on X = Spec K[x 1 , . . . x n ] where x i := (xi1 , . . . , xiki ) are multivariables and x αi i = aik a xi1i1 · . . . · xiki i ), such that a1 := |α1 | < . . . < an := |αn |. This is a generalization of Examples 1.6, and 4.67. Then R1 = OX [f t]. The order of f at the origin 0 is a1 . Thus the first maximal contact at 0 is x 1 ∈ Da1 −1 (f ) ⊂ x 1 + m2p . Moreover H1 := V (x 1 ) splits in X, and Dx 1 splits. Thus, by Lemma 4.32, R2 can be written in the split form as: R2 = Cx 1 t 1/a1 (R1 ) = OX [x 1 t 1/a1 , (x α2 2 + . . . + x αnn )t],

.

Similarly by the induction, and Lemma 4.43, Ri can be written in the split form: α

i+1 + . . . + x αnn )t], Ri = Cx i t 1/ai (Ri−1 ) = OX [x 1 t 1/a1 , . . . , x i t 1/ai , (x i+1

.

where x i ∈ Dai −1 (x αi i + . . . + x αnn ) = THaii−1 (Ri−1 )

.

is a nested maximal contact of Ri−1 at Hi−1 = V (x 1 , . . . , x i−1 ), and Hi = V (x 1 , . . . , x i ) splits. So the maximal admissible center is A = Rn+1 = OX [x 1 t 1/a1 , . . . , x i t 1/an ]int ,

.

and invp (f ) = (a1 , . . . , a1 , a2 , . . . , a2 , . . . , an , . . . , an ),

.

where each ai is repeated ki times.

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Then the extended center is Aext = OX [t −1/wA , x 1 t 1/a1 , . . . , x i t 1/an ],

.

where wA := lcm(a1 , . . . , an ) = w1 a1 = . . . = wn an .

.

The full cobordant blow-up σ : B → X at A is an affine space B = Spec(K[t −1 , x 1 t w1 , . . . , x n t wn ]),

.

with the coordinates (t −1 , x 1 , . . . , x n ), where x i := x i t wi . The function f t wA = σ c (f ) ∈ OB is the controlled transform of f . We write σ c (f ) = t wA (x α1 1 + . . . + x αnn ) = t wA (x 1 t −w1 )α1 + . . . + (x n t −wn )αn ) =

.

= (x 1 )α1 + . . . + (x n )αn On B+  V (x 1 , . . . , x n ) the invariant clearly drops. In the particular situation from Example 1.6 we have inv0 (x1a1 + . . . + xnan ) = (a1 , . . . , an ).

.

In this case the cobordant blow-up at OX [t −1/wA , x1 t w1 , . . . , xn t wn ] resolves singularity. The observation about resolution is also valid under some mild assumptions in characteristic p , but different arguments are used.(see Example 5.9 and Proposition 5.10). In particular, it is true when the singular locus is equal to V (x1 , . . . , xn ). In such a case we remove the singular locus Vert(B) = V (x1 , . . . , xn ) from the hypersurface described by the controlled transform σ c (f ) = (x1 )a1 + . . . + (xn )an . This example as many others shows the relation between the center of the full cobordant blow-up and its vertex. (see Example 5.9 and Proposition 5.10) Example 4.87 Let f = (x1 + x2 )(x3 + x43 ), where x1 , x2 , x3 are divisorial at the origin and x4 is free. Then R1 = OX [f t]. The order of f at the origin p = 0 is 2, and thus ordp (R1 ) = 2. Computing D(f ) = ((x1 + x2 ), (x3 + x43 ))

.

we see that the linear parts are combinations of the divisorial unknowns x1 , x2 , x3 . Thus the maximal contact is x 1 := (x1 , x2 , x3 ) in gradation t 1/2 . The subvariety of maximal contact H1 := V (x 1 ) splits in X. Write the generator f in the coefficient form with respect to

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x 1 t 1/2 := (x1 t 1/2 , x2 t 1/2 , x3 t 1/2 ) as in Lemma 4.32: f = (x1 +x2 )(x3 +x43 )t = (x1 t 1/2 x2 t 1/2 +x3 t 1/2 x1 t 1/2 )+(x2 t 1/2 )x43 t 1/2 +(x1 t 1/2 )x43 t 1/2

.

we find the only coefficient cx1 t 1/2 = cx2 t 1/2 = x43 t 1/2 . The coefficient ideal, by Lemma 4.32, can be written in the split form as R2 = Cx 1 t 1/2 R1 = OH [x43 t] = OX [((x1 , x2 , x3 )t 1/2 )Int , x43 t 1/2 ].

.

The nested order of R2 is equal to ordp (x43 t 1/2 ) = 6. Its maximal contact is t −2/6 Dx 2 (x43 t 1/2 ) = x4 t 1/6 . Thus for x 2 := x4 we obtain the center A = R3 , where 4

R3 = Cx 2 t 1/6 R2 = Cx 2 t 1/6 (OX [((x1 , x2 , x3 )t 1/2 )Int , x43 t 1/2 ]) =

.

=OX [(x1 , x2 , x3 )t 1/2 , x4 t 1/6 ]int . Since x1 , x2 , x3 are divisorial, and x4 is free, we have invp (f ) = inv(J ) = (2+ , 2+ , 2+ , 6).

.

The extended center is Aext = OX [t −1/6 , (x1 , x2 , x3 )t 1/2 , x43 t 1/6 ],

.

where wA = lcm(2, 6) = 6. The controlled transform of f under the full cobordant blow-up B = OX [t −1 , x1 t 3 , x2 t 3 , x3 t 3 , x4 t],

.

is exactly f = σ c (f ) = t −6 f = (x1 + x2 )(x3 + (x4 )3 ) = uv,

.

where u := x1 + x2 , and v = x3 + (x4 )3 . At the vertex V (x1 , x2 , x3 , x4 ), the coordinates x1 , x2 , x3 are divisorial, and the function f has the form f = (x1 + x2 )(x3 + (x4 )3 ),

.

which is exactly the same as f .

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On (V (f ) ∩ V (x3 ))  V (x1 , x2 ), where x3 is divisorial and (x3 , x4 , u) is a partial coordinate system compatible with the induced divisor E , the equation is f = u(x3 + (x4 )3 ),

.

where u, x4 are free maximal contacts. with the associated center equal to ((u, x3 )t 1/2 , x4 t 1/6 ),

.

(written using a shorter notation as in Sect. 4.1.4) and the invariant (2, 2+ , 6). On the other hand on (V (f ) ∩ V (x1 , x2 ))  V (x3 , x4 ), where x1 , x2 are divisorial and (x1 , x2 , v) is a partial coordinate system compatible with E , we can write f = (x1 + x2 )v.

.

The center is (v, x1 , x2 )t 1/2 ,

.

with the invariant (2, 2+ , 2+ ).

.

Finally the invariant on V (f )  V (x1 , x2 )  V (x3 ) is just (2, 2) with the center (u, v)t 1/2 corresponding to f = uv. In particular, the maximal invariant drops on B+ = B  V (x1 , x2 , x3 , x4 ).

5

Examples of the Resolution in Positive Characteristic

5.1

Negative Examples in Positive Characteristic

In this section, we will demonstrate some problems of the resolution invariant .invp (I) in positive characteristic. In characteristic zero the Rees center .A associated with .invp (I) at a point p is unique and determines the associated blow-up. Moreover, the vanishing locus of the center is given by the set of points where the invariant is constant. This is no longer the case in nonzero characteristic. For simplicity, we assume that K is an algebraically closed field of characteristic p. The following examples are variations of the examples from [46].

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Example 5.1 Let x p + y p z ∈ K[x, y, x, w]

.

defines the hypersurface X in .A4 = Spec(K[x, y, x, w]). Then at the origin, there is a group of automorphisms isomorphic to .A1K preserving X. With any .a ∈ K we associate the action a(x, y, z, w) = (x + aty, y, z − a p t p , w)

.

This determines a family of the associated centers ((x + awy)t 1/p , yt 1/(p+1) , (z − a p w p )t 1/(p+1 ))

.

parametrized by .a ∈ K and associated with the invariant .invp (I) = (p, p + 1, p + 1) at the origin. One can easily see that all those centers are different. Moreover their vanishing loci V (x + awy, y, z − a p w p ) = V (x, y, z − a p w p )

.

are different. All the coordinate systems defining the centers differ by the action of the additive group .Ga = K, and thus none can serve as the canonical center at the origin. Observe that the center .(xt 1/p , yt 1/p ) has good resolution properties, but it is not maximal admissible for the points in .V (x, y). A similar principle is used in the next example. Example 5.2 Consider the equation p

p

x p + y p z + (z − t1 )r · . . . · (z − tk )r ∈ K[x, y, z, t1 , . . . , tk ]

.

where .r ≥ p + 1 and r is not divisible by p. In this case, we have k associated distinct centers, which are exactly p

((x + ti y)p , y p+1 , (z − ti )p+1 ).

.

Moreover, there is a group of permutations .Σn acting on them. Note that here the p p associated centers depend upon the term .(z − t1 )r · . . . · (z − tk )r of arbitrarily large order. p Thus the term of large order affects the weighted initial form .(x + ti y)p + y p (z − ti ). Moreover there is no single canonical center improving singularities.

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Another problem is related to the locus of the points with a fixed invariant. The smooth center is determined in characteristic zero by the invariant .invp (I). This is not the case with positive characteristic. Example 5.3 Let .A2n+k+1 = Spec(K[x1 , . . . , xn , y, z1 , . . . , zn , t1 , . . . , tk ]) be the ambient variety, and .An = V (x1 , . . . , xn , y, t1 , . . . , tk ) = Spec(K[z1 , . . . , zn ] be its affine subspace. The following example shows that any closed subset Z of .An can be realized as the set of the closed points where .invp (I) is constant for a certain ideal .I associated with Z. Let .Z ⊂ An = Spec(K[z1 , . . . , zn ]) be the closed subset defined by an arbitrary set of generators fi (z1 , . . . , zn ) ∈ K[z1 , . . . , zn ]

.

for .i = 1, . . . k. Consider the affine space A2n+k+1 = Spec(K[x1 , . . . , xn , y, z1 , . . . , zn , t1 , . . . , tk ]),

.

and let the subvariety X be defined by the set of equations p

x1 + y p z1 = 0

.

... p

xn + y p zn = 0 p+1

t1

+ f1s (z1 , . . . , zn ) = 0

... p+1

tk

+ fks (z1 , . . . , zn ) = 0,

where .s ≥ p + 1 is any integer. Denote by .0 ∈ A2n+1 its origin. The maximal invariant is equal to inv0 (I) = (p, . . . , p, p + 1, . . . , p + 1),

.

with p occurring n times and .p + 1 repeated .n + k + 1 times. The corresponding locus S = {q ∈ A2n+k+1 | inv0 (I) = invp (I)}

.

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is given exactly by the set S = Z = V (x1 , . . . , xn , y, t1 , . . . , tk , f1 , . . . , fn )

.

= V (f1 , . . . , fn ) ⊂ Spec(K[z1 , . . . , zn ]) The essential property in this example is that .fi (a) = 0, ti = 0 for any .a ∈ Z so that p+1

ordp (ti

.

+ fis ) ≥ p + 1.

For a closed point .a = (a1 , . . . , an ) ∈ An consider its Frobenius inverse image 1/p

1/p

a 1/p = (a1 , . . . , an ) ∈ An .

.

Then at every point .a ∈ Z the associated Rees center is given by 1/p

Aa := ((x + ai

.

y)t 1/p ,

yt 1/(p+1) ,

(z − ai )t 1/(p+1) ,

ti · t 1/(p+1) ),

where .V (Aa ) = {a} ⊂ Z. Thus the corresponding centers form an infinite family parametrized by the closed points .a ∈ Z. It is not clear what center shall be blown up and whether, in general, there exists a weighted center that improves the singularity at a given point .a ∈ Z. Again the initial form depends upon the terms .fis of an arbitrarily large order. Remark 5.4 What is common to many problems in positive characteristic is that the centers of the blow-ups, the weights, and the coordinates depend heavily on specific higher-order terms which do not occur in the weighted initial forms and are in general hard to identify. These terms are not canonical, and not easily accessible to the differential methods likewise the algebraic methods typically relying on the initial forms. On the other hand, the weighted initial forms do not give enough hints about the plausible centers of blow-ups.

5.2

Deformation to the Weighted Normal Cone

5.2.1 Weighted Normal Cone Even though there is no general method of resolution of singularities by the weighted or more general centers, positive characteristic resolution of singularities by weighted centers is still possible for certain singularities. Let W be a regular affine scheme. For simplicity consider .X = Spec(O(W )[x1 , . . . , xk ]). Consider a center .A = (x1 t 1/a1 , . . . , xk t 1/ak ) on k with .V (A) = W and let .ν = ν be the associated monomial valuation such that .X = A A W .ν(xi ) = 1/ai as in Lemma 3.5. Let .I be an ideal of .S := O(W )[x1 , . . . , xk ] contained in

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(x, . . . , xk ). Then with any polynomial .f =

.

.



cα x α ∈ S we can associate its initial form



inν (f ) =:

cα x α .

ν(x α )=ν(f )

The corresponding ideal .

inν (I) = {inν (f ) | f ∈ I}

will be called the ideal of the initial forms. The subscheme CX/V (A) := V (inν (I))

.

of .X = AkW shall be called the weighted normal cone of X at .V (A). This definition can be given in more general terms as a subscheme of a normal bundle to the regular subvariety .V (A) in [49], and [36]. It plays an important role in the resolution by weighted centers. Theorem 5.5 ([36, 49]) Let X a regular scheme, and .σ : B → X be the cobordant blowup of a weighted center .A, with .Z := V (A) ⊂ Y . Then the exceptional divisor .VB (t −1 ) of B is isomorphic to the weighted normal bundle .NX/V (A) of X at the center .V (A): V (t −1 ) = Spec(OB /(t −1 )) = Spec(OX [t −1 , x1 t w1 , . . . , xk t wk ]/(t −1 ) =

.

= Spec(OV (A) [x1 t w1 , . . . , xk t wk ]) = NX/V (A) X. ♣

.

Lemma 5.6 ([49]) With the previous assumptions and notation. Let .I ⊂ OX be an ideal sheaf on X, such that .V (I) contains .V (A). Let .σ s (I) ⊂ OB be the strict transform of .I. Then the natural isomorphism .OB /(t −1 ) → OV (A) [x1 t w1 , . . . , xk t wk ]) takes .σ s (I)|V (t −1 ) to .inν (I). Thus let .Y ⊂ X be a subscheme of a regular scheme X, containing .V (A). If .σY : Y → Y is the induced morphism, where .Y is the strict transform of Y , then the .σY−1 (V (A)) = VB (t −1 ) ∩ Y is isomorphic to the weighted normal cone .CY /V (A) . Proof For simplicity assume that .X = Spec(O(W )[x1 , . . . , xn ]), where W is affine and regular. Thus the subscheme .Y ⊂ X defined by fi (x1 , . . . xn ) = Fi (x1 , . . . xn ) + Hi ,

.

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where each .Fi (x1 , . . . xn ) is quasihomogeneous for .ν, and such that .ν(Hi ) > ν(Fi ) = ν(fi ), for .i = 1, . . . , r. The full cobordant blow-up .σ : B → X of .A transforms .fi into σ ∗ (fi ) = fi (t −w1 x1 , . . . t −wn xn ) = t −ν(fi ) Fi (x1 , . . . , xn ) + t −ν(Hi ) H (x1 , . . . , xn ) =

.

t −ν(fi ) (Fi (x1 , . . . , xn ) + t −ν(Hi )+ν(fi ) Hi (x1 , . . . , xn )),

.

where H (x1 , . . . , xn ) := t ν(Hi ) Hi (t −w1 x1 , . . . t −wn xn )

.

Thus the ideal of the strict transform .Y of Y is generated by the functions Fi (x1 , . . . , xn ) + t −ν(Hi )+ν(fi ) Hi (x1 , . . . , xn ).

.

Its restriction .Y|V = Y ∩ V (t −1 ) to the exceptional divisor .V (t −1 ) gives exactly the (t −1 ) .♣ equations .Fi (x1 , . . . , xn ) of the weighted normal cone .CY /V (A) .

5.2.2 Deformation to the Weighted Normal Cone In the joint paper [36] of Quek and Rydh made the following observation (in the context of stacks) (See also [49]). Corollary 5.7 ([36]) .X = AnK be an affine space over a field K. Let .σ : B → X be a full cobordant blow-up of a weighted center Aext = OX [t −1 , x1 t 1/a1 , . . . , xk x1 t 1/ak ]

.

on a regular variety X. The natural projection .π : B → A1K = Spec K[t −1 ] is a deformation to the weighted normal cone. Indeed π −1 (A1  {0}) = B  V (t −1 ) X × (A1  {0}

.

is a trivial X-bundle, with the fibers isomorphic to X. On the other hand the special fiber π −1 ({0}) = V (t −1 ) is isomorphic to the weighted normal bundle .NX/V (A) of X at the center .V (A). Let .Y ⊃ V (A) be a subvariety containing .V (A), and .σY : Y → Y be the induced morphism, where .Y ⊂ B is the strict transform of Y . Let .πY : Y → A1K be the restriction of .π to .Y ⊂ B. Then .πY−1 (Gm ) Y ×Gm is a trivial Y -bundle, while .πY−1 (0) CY /V (A) .

.

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5.3

Characteristic-Free Resolution of Quasihomogeneous Singularities

Let W be a regular scheme and .I be an ideal of .AnW = SpecW (OW [x1 , . . . , xk ]). Consider a center .A = (x1 t 1/a1 , . . . , xk t 1/ak ) on .An , and let .ν = νA be the associated valuation such that .ν(xi ) = 1/ai . We shall call the ideal .I quasi-homogenous with respect to .ν if .I = inν (I), so .I is generated by quasi-homogenous forms with respect to .ν. The following phenomenon can be observed in many examples. Lemma 5.8 ([49]) Let .I be a quasi-homogenous ideal of X = AnW = SpecW (OW [x1 , . . . , xk ]).

.

The full cobordant blow-up .B → X = AnW of .A = (x1 t 1/a1 , . . . , xn t 1/an ) is isomorphic to 1 n B An+1 W = AW ×W AW = SpecW (OW [y, x1 , . . . , xk ]).

.

Let .π : A1W ×W AnW → AnW denote the standard projection. Under these identifications the strict transform .σ s (I) corresponds to .π −1 (I)OAnW , and the vertex .Vert(B) = B  B+ corresponds to .VAn+1 (x1 , . . . , xn ). W

Proof B = Spec(OX [t −1 , x1 t w1 , . . . , xn t wn ]) =

.

Spec(OW [x1 , . . . , xn ][t −1 , x1 t w1 , . . . , xr t wr ]) = = Spec(OW [t −1 , x1 t w1 , . . . , xr t wr ]) An+1 W The coordinates on .An+1 are given by .t −1 , x1 , . . . , xn , where .xi = t wi xi , for .i ≤ r and k .x = xi , for .i > r. Then for any quasihomogeneous .F ∈ I we have i σ s (F ) = σ c (F ) = t ν(F ) (F (t −w1 x1 , . . . , t −wr xr , xr+1 , . . . , xn )) = F (x1 , . . . , xn ).

.

The coordinates on .An+1 are given by .t −1 , x1 , . . . , xn , where .xi = t wi xi , for .i ≤ r and k .x = xi , for .i > r. Then for any quasihomogeneous .F ∈ I we have i σ c (F ) = t ν(F ) (F (t −w1 x1 , . . . , t −wr xr , xr+1 , . . . , xn )) = F (x1 , . . . , xn ).

.

♣

.

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The following example shows how to resolve singularities using the above Lemma. It is a generalization of Example 1.6. Example 5.9 Let X be defined by .f = x1a1 + . . . + xnan in .AnW . Consider the center 1/a1 , . . . , x t 1/an ) on .An , computed in characteristic zero in Example 4.86, .A = (x1 t n W and let .ν = νA be the associated valuation. Let .X = CX/V (A) be a subvariety defined by quasihomogeneous equations .F1 , . . . , Fr in .AnW and having singularities at .V (A). Then n+1 .B A W is defined by the same equations. Consequently, when removing the singular locus A1W × V (A) V (σ c (A)) = Vert(B)

.

the strict transform of X in B+ = B  V (σ c (A)) A1W × (AnW  V (A))

.

becomes isomorphic to .A1W × (X  V (A)), and thus is regular with a nontrivial torus action.

5.4

Characteristic-Free Resolution of Isolated Singularities

Lemma 5.8, and the result below is an illustration of two principles: (1) Upon the right choice of the center the worst singularities on the full cobordant blowup B are at the vertex .Vert(B) = B  B+ . (2) The behavior around the vertex .Vert(B) of the full cobordant blow-up B is to some extent controlled by the weighted normal cone to the center .V (A) of the blow-up as in Lemma 5.6. A similar principle to (1) was stated in characteristic zero for maximal admissible centers in Sect. 4.4.18. The following proposition generalizes Example 1.6. It is a particular case of a more general theorem proven in [49]. Definition 5.10 ([49]) Let W be a regular universally catenary affine scheme, and let 1/a1 , . . . , x t 1/ak ) be a center in .X = An = Spec(O [x , . . . , x ])), and let .A = (x1 t k W 1 k W .ν = νA be the associated valuation as in Lemma 3.5. Let .Y ⊂ X be a closed irreducible subscheme of X containing .V (A). Assume that the singular loci .Sing(Y ) of Y , and .Sing(CY /V (A) ) of .CY /V (A) are contained in .V (A). Then the cobordant blow-up .B+ → X at .A resolves the singularity Y , so .σBs + (Y ) ⊂ B+ is regular.

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Proof Let .r = codimY (X). Then for the morphism .σ− : B− = X × Gm → X, the inverse image .σ−−1 (Y ) is irreducible of codimension r. So it is its closure .Y := σ −1 (Y ), which is the strict transform of Y . Thus .codimB (Y ) = r. Note that by the definition of the strict transform, .t −1 is not a zero divisor of .OB /IY . Then, by the Krull Hauptidealsatz, we have that each component of .Y ∩ V (t −1 ) is of codimension 1 in .Y , and the codimension .r + 1 in B. We conclude that the restriction .Y|V = Y ∩ V (t −1 ) of .Y to the exceptional (t −1 )

divisor .V (t −1 ) is of codimension r in .V (t −1 ). By Lemma 5.6, the restriction .Y ∩ V (t −1 ) is isomorphic to the cone .CY /V (A) in .NX/V (A) X and thus is regular outside of .V (A). Consequently Y ∩ V (t −1 ) ∩ B+ = Y ∩ V (t −1 )  V (σ c (A))

.

is locally defined by r local parameters .u1 , . . . , ur on .V (t −1 ) ∩ B+ , where r = codimV (t −1 ) (Y|V ). (t −1 )

.

These local parameters lift to local parameters .u 1 , . . . , u r ∈ IY on .B+ , defining locally a regular subscheme .Y = V (u 1 , . . . , u r ) ⊂ B+ of codimension r and containing .Y . This implies that the subscheme .Y ∩ B+ = Y is regular in a neighborhood of V (t −1 )  V (x1 , . . . , xk ) ⊂ B+ = B  V (σ c (A)).

.

But since B− = B  V (t −1 ) = AnW ×Z Gm ,

.

we infer that Y  V (σ c (A))  V (t −1 ) = (Y  V (A)) ×Z Gm

.

is also regular. Thus the singularity locus of .Y on B is given by V (σ c (A)) = V (x1 , . . . , xn ),

.

and the cobordant blow-up .B+ = B  V (x1 , . . . , xn ) → X defines a regular subscheme .Y ∩ B+ ⊂ B+ . .♣

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307

Generalized Blow-Ups and Cox Rings

Weighted blow-ups generalize regular blow-ups allowing for much greater flexibility for the resolution algorithms. They also eliminate many technical problems associated with regular centers. Moreover, there are many nontrivial examples or classes of singularities when weighted centers give a visible improvement or even immediate resolution of singularities (up to a torus action). On the other hand, the pathological examples in positive characteristic suggest that more general centers may be handy. Abramovich-Quek, in their recent work [2], introduced so-called multiple weighted blow-ups to enlarge the pool of the centers to locally monomial ideals. They used a construction of Satriano for toroidal morphism, which locally reduces the transformation to a toric language, represented in the combinatorial language of locally toric smooth Artin stacks. Shortly after, this technique was further generalized and put into a more geometric context of the relative Cox rings in [50].

6.1

Cox Rings

Recall the Cox ring construction, originally introduced by Cox in the context of toric varieties and generalized by Hu-Keel. Definition 6.1 Let X be a normal toric variety over a field K with a finitely generated Weil divisors class group .Cl(X). By the Cox ring of X, we mean C(X) :=



H 0 (X, OX (D)),

.

D∈Cl(X)

where .Cl(X) is the Weil divisor class group, and H 0 (X, OX (D)) = {f ∈ K(X) | div(f ) + D ≥ 0}.

.

The action of torus .T = Spec K[Cl(X)] naturally occurs in the construction and is determined by the .Cl(X)-gradation. The Cox formula extends the presentation of the coordinate ring of the projective space n .X = P , namely K[x0 , . . . , xn ] =



.

n∈Z

H 0 (X, OX (n)).

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J. Włodarczyk

The projective space .X = Pn itself can be viewed as the geometric quotient X = Pn = (An+1  {0})/Gm .

.

The space X := Spec(C(X)) = Spec(

.

 (H 0 (X, OX (n))) = Spec(K[x0 , . . . , xn ]) = An+1 n∈Z

is called the coordinate space of X. On the other hand the geometric quotient (An+1  {0}) → Pn = (An+1  {0})/Gm

.

is defined by the .OX (1)-bundle over X with removed zero section, which can be generalized as the characteristic space of X: 

Xˆ = SpecX (

OX (D)) → X

.

D∈Cl(X)

introduced in [8]. The characteristic space .Xˆ comes with the natural embedding .Xˆ → X into the coordinate space: 

X = Spec(

.

H 0 (X, OX (D)).

D∈Cl(X)

In particular, for .X = Pn we get n+1 Xˆ = An+1 K  {0} → X = AK .

.

6.2

Relative Cox Rings

Let .π : Y → X be a proper birational morphism of normal integral schemes. Denote by Cl(Y /X) ⊂ Div(Y ) the free group generated by the images of the exceptional irreducible Weil divisors .Ei of .π in .Cl(Y /X). It can be identified with the kernel of the surjective morphism .π∗ : Cl(Y ) → Cl(X).

.

Definition 6.2 By the relative Cox ring w mean the sheaf of graded .OY -algebras CY /X =



.

E∈Cl(Y /X)

CE :=

 E∈Cl(Y /X)

OY (E),

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graded by .Cl(Y /X), where .CE := OY (E) for OY (E)(U ) = {f ∈ K(Y ) | (divY (f ) + E)|U ≥ 0} ⊂ K(Y ) = K(X).

.

Note the .C0 = OY . We introduce the dummy variables .t = (t1 , . . . , tk ) so that .Ei  corresponds to .ti−1 and .E = ai Ei → tE = t1−a1 · . . . · tk−ak , and put .OY (E) = OY (α), for .α = (−a1 , . . . , −ak ). We can write 

CY /X =

.

CE t E =



OY (α) · t1a1 · . . . · tkak

α∈Zk

E∈Cl(Y /X)

6.2.1 Exceptional Valuations By the exceptional valuations of .π : Y → X, we shall mean the valuations .ν1 , . . . , νk of the quotient field .K(X) = K(Y ) associated with the generic points of the exceptional divisors .E1 , . . . , Ek of .π . These valuations define ideals .Iν,a,X on X for .a ∈ Z, such that .Iν,a,X ⊂ OX is generated by the functions .f ∈ OX , with .ν(f ) ≥ a. In particular .Iν,a = OX if .a ≤ 0. Lemma 6.3 ([50]) Let .E =



ni Ei correspond to .t1−n1 · . . . · tk−n1 . Then

(1) .π∗ (OY (Ei )) = OX . (2) If all .ni ≥ 0 then .π∗ (OY (E)) = OX . (3) If there is .ni < 0, then π∗ (OY (E)) =



.

ni