Del Pezzo and K3 Surfaces

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Ivan Cherednik, Peter J. Forrester and Denis Uglov, Quantum many-body problems and representation theory, 1998 Masako Takahashi, Mitsuhiro Okada and Mariangiola Dezani-Ciancaglini (Eds.), Theories of types and proofs, 1998 Tomotada Ohtsuki, Combinatorial quantum method in 3-dimensional topology, 1999 Atsushi Matsuo and Kiyokazu Nagatomo, Axioms for a vertex algebra and the locality of quantum fields, 1999 Daryl Cooper, Craig D. Hodgson and Steven P. Kerckhoff, Three-dimensional orbifolds and cone-manifolds, 2000 Kong De-xing, Cauchy problem for quasilinear hyperbolic systems, 2000 Vladimir Georgiev, Semilinear hyperbolic equations, 2000 Eric M. Opdam, Lecture notes on Dunkl operators for real and complex reflection groups, 2000 \cdots \cdots \cdots \cdots \cdots

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Del Pezzo and K3 surfaces

Valery Alexeev and Viacheslav V. Nikulin

UNIVERSITY OF GEORGIA, ATHENS, GA 30602, USA E-mail address: [email protected]. edu DEPT. OF PURE MATH., UNIVER. OF LIVERPOOL,XW LIVERPOOL L69 3BX, UK; STEKLOV MATH. INST., UL. GUBKINA 8, Moscow 117966, GSP-I, RUSSIA E-mail address: vnikulin@liv. ac. uk, vvnikulin@list. ru

MSJ Memoirs

Mathematical Society of Japan

Del Pezzo and K3 surfaces

Valery Alexeev and Viacheslav V. Nikulin

Volume 15

2006

Author Valery Alexeev

Viacheslav V. Nikulin Professor Dept. of Pure Math. University of Liverpool XW Liverpool L69 3BX, UK [email protected] Steklov Math. Inst. ul. Gubkina 8 Moscow 117966, GSP- , Russia [email protected]

Professor University of Georgia Athens GA 30602, USA

[email protected]

$I$

AMS Subject Classifications: Primary 14J25,14J26,14J28; Secondary 14J17,14J45,14J50, 11E12, 11F22, 22E40

MSJ Memoirs This monograph series is intended to publish lecture notes, graduate textbooks and long research papers in pure and applied mathematics. Each volume should be an integrated monograph. Proceedings of conferences or collections of independent papers are not accepted. Articles for the series can be submitted to one of the editors in the form of hard copy. When the article is accepted, the author(s) is (are) requested to send a camera-ready manuscript. limited to contributions by MSJ members $*$

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Editorial Board Fukaya, Kenji Funaki, Tadahisa Kashiwara, Masaki Kobayashi, Ryoichi Kuniba, Atsuo Maeda, Yoshiaki Miwa, Tetsuji Miyaoka, Yoichi (Chief Editor/Managing Editor) Ohta, Masami Okamoto, Kazuo Okamoto, Hisashi Ozawa, Tohru Taniguchi, Setsuo Tsuboi, Takashi

Uzawa, Tohru

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Orders from all countries except Japan are to be sent to: JAPAN PUBL. TRADING CO.,LTD MARUZEN CO., LTD International Division P.O.Box 5030 Tokyo International P.O.BOX 5050 Tokyo 100-3191, Japan Tokyo International Tokyo 100-3191, Japan FAX81332920410 E-mail: [email protected] Tel: 81332733234 FAX: 81332789256 E-mail: [email protected] vi

Foreword

The classification of non-singular del Pezzo surfaces is classical and well known. One of the striking features of the theory is that their geometry is govemed by the Weyl groups of finite root systems. In this book, we extend this classical theory to the del Pezzo surfaces of index one or two by completely determining the weighted graphs of the exceptional curves which appear in their standard resolutions (the right resolutions) of the singularities. Our final result is quite similar to the classical case, and everything again is govemed by finite root systems. However, the arguments used here are not so classical, relying essentially on the theory of K3 surfaces and of reflection groups in hyperbolic spaces attached to the Picard lattice of K3 surfaces. This book grew up from ajoint paper [AN88] in Russian by the present two authors. The authors hope that this enlarged and self-contained English exposition will be accessible to wider audience and will well serve as an introduction to the theoly of del Pezzo surfaces and K3 surfaces $\log$

vi

Contents

Foreword

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

List of Tables

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

List of Notations

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

vii xi

xii

Introduction\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots 0.1. Historical remarks and our main principle of classification of $\log$ del Pezzo surfaces of index 0.2. Classification of log del Pezzo surfaces of index and K3

1

surfaces Final classification results for log del Pezzo surfaces of index

4

$\leq 2$

\cdots

\cdots

1

$\leq 2$

\cdots

0.3.

\cdots

\cdots

\cdots

5

$\leq 2$ \cdots

Chapter 1.

\cdots

\cdots

del Pezzo surfaces of index and Smooth Divisor Theorem 12 Basic definitions and notation.\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots 12 terminal singularities of index 2. \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots 13 Basic facts about del Pezzo surfaces \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots14 Smooth Divisor Theorem 15 Reduction to DPN surfaces of elliptic type. \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots18 $\leq 2$

${\rm Log}$

\cdots

1.1. 1.2. 1.3. 1.4. 1.5.

\cdots

\cdots

\cdots$\cdots\cdots$

\cdots

${\rm Log}$

$\log$

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

Chapter 2.

2. 1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7.

General Theory of DPN surfaces and K3 surfaces with non-symplectic involution \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots20 General remarks 20 Reminder of basic facts about K3 surfaces. 22 The lattice , and the main invariants 29 and $Y$ \cdots \cdots \cdots \cdots \cdots \cdots \cdots32 Exceptional curves on The root invariant of a pair 39 Finding the locus 43 Conditions for the existence of root invariants 44 \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

$S$

$(r, a, \delta),$

$(k, g, \delta)$

\cdots

\cdots

$(X, \theta)$

$(X, \theta)$

\cdots

\cdots

\cdots

$X^{\theta}$

\cdots

\cdots

\cdots

\cdots \cdots \cdots \cdots \cdots \cdots

ix

X

CONTENTS

2.8. Three types of non-symplectic involutions of K3 surfaces

\cdots

Chapter 3. DPN surfaces of elliptic type 3.1. Fundamental chambers of $W^{(2,4)}(S)$ for elliptic type T.l. Table 1 for 3.2. Root invariants, and subsystems of roots in elliptic case of elliptic 3.3. Classification of non-symplectic involutions type of K3 surfaces T.2. Table2 3.4. Proof of Classification Theorem 3.6 3.5. Classification of DPN surfaces of elliptic type T.3. Table3 3.6. Application: On classification of plane sextics with simple singularities T.4. Table 4 \cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\cdots$

\cdots \cdots \cdots \cdots \cdots \cdots

48 50 50 52

$\Delta^{(4)}(\mathcal{M}^{(2)})$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

56

$(X, \theta)$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

Chapter 4.

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

58 64 68 89 93

100 103

del Pezzo surfaces of index Classification of and applications 104 4.1. Classification of log del Pezzo surfaces of index 104 4.2. Example: Enumeration of all possible types for $N=20$ 112 4.3. Application: Minimal projective compactifications of affine del Pezzo surfaces of surfaces in by relatively minimal index 114 4.4. Dimension of the moduli space 115 4.5. Some open questions 116 $\leq 2$

$\log$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\leq 2$

\cdots

\cdots

\cdots

\cdots

\cdots

$\mathbb{P}^{2}$

$\log$

$\leq 2$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

Appendix 117 A. 1. Integral symmetric bilinear forms. Elements of the discriminant forms technique 117 A.2. Classification of main invariants and their geometric interpretation 122 A.3. The analogue of Witt’s theorem for 2-elementary finite forms 126 A.4. Calculations of fundamental chambers 128 \cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

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\cdots

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\cdots

References Index

\cdots

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\cdots

\cdots

\cdots

\cdots

142

\cdots

\cdots

147

List of Notations

lii

the characteristic element (if$ $R(X, \theta)$

$a$

$A_{n}$

$b$

\cdots

\cdots

\alpha=0$ )

\cdots

one of main invariants:

$(S^{*}/S)\cong(\mathbb{Z}/2\mathbb{Z})^{a}$

$b_{M}$

$\det(M)$

Duv $()$ $D_{n}$

$E_{n}$

, of $Z,$

$Y,$

.. .

$(X, \theta)$

Dynkin diagram, Du Val singularity, root system of type a finite symmetric bilinear form one of elementary finite symmetric bilinear forms the discriminant bilinear form of a lattice $M$ the determinant of a lattice $M$ the Du Val part of Dynkin diagram, Du Val singularity, root system of type Dynkin diagram, Du Val singularity, root system of type genus of a non-singular curve in $|-2K_{Z}|$ or of moving part of $|-2K_{Y}|$ , maximal genus of a component of $g=$ $A_{n}.$

\cdots

$b_{\theta}^{(p)}(p^{k})$

of the root invariant

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$(\cdot)$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots \cdots \cdots

$D_{n}$

\cdot \cdot

$E_{n}$

\cdot \cdot

$g$

41 30

5 117 118 118 118 61 5 5

$X^{\theta},$

$g$

$(22-r-a)/2$ a part (kemel) of the root invariant $R(X, \theta)$ hyperplane orthogonal to of hyperbolic space half-space orthogonal to of hyperbolic space number of exceptional $-4$ curves on $Y$ ; number of double transparent vertices of the graph ; number of genus curves \cdots \cdots \cdots

$H$ $\mathcal{H}_{\beta}$

\cdots

\cdots

\cdots

$\beta$

\cdots

$\mathcal{H}_{\beta}^{+}$

\cdots

$K_{H}$ $K_{n}$

$K_{\theta}^{(p)}(p^{k})$

${\rm Log}(\cdot)$

$l(\mathfrak{A})$

\cdots

of $X^{\theta};k=(r-a)/2$ altemative main invariants of $Z,$ $Y,$ a part (the root lattice $K$ or $K(2)$ ) of the root invariant $R(X, \theta)$ the overlattice of $K$ related to the root invariant $R(X, \theta)$ terminal singularities of index 2 one of (with rank 1) elementary -adic lattices the logarithmic part of the minimal number of generators of a finite Abelian group \cdots

$K$

\cdots

\cdots

\cdots

$(X, \theta)$

\cdots

\cdots

\cdots

\cdots

$\log$

\cdots

$p$

$(\cdot)$

\cdots

\cdots

\cdots

$\cdots\cdots$

\cdots

\cdots

\cdots

\cdots

$\cdots$

\cdots

\cdots

$\mathfrak{A}$

xii

24 24

$0$

$\Gamma$

$(k, g, \delta)$

\cdots

$\beta$

\cdots

$k$

\cdots

31 41

31 32 40 44 18 118 61 45

xiii

LIST OF NOTATIONS $L_{K3}$ $\mathcal{L}(X)$

$M(a)$ $M$

the abstract lattice isomorphic to for a K3 surface X , the hyperbolic space of a surface X $H^{2}(X, \mathbb{Z})$

$\mathcal{L}(X)=V^{+}(S_{X})/\mathbb{R}^{+}$

\cdots

multiplication by of the form of a lattice $M$ 117 an arbitrary lattice usually; in Sect. 2.7, the lattice related to 117 the root invariant and defined in (57) $\mathcal{M}(X)=$ NEF the projectivization of the $X$ 24 nef cone of a surface a fundamental chamber of $W^{(2)}(S)$ in 36 a fundamental chamber of $W^{(2,4)}(S)$ in 36 a fundamental chamber of in 37 moduli space of K3 with condition $M$ on Picard lattice. . 29 moduli space of K3 with condition $M\subset L_{K3}$ on Picard lattice 28 with the main invariants 30 moduli space of moduli space of 30 with the main invariant $Y,$ labels main invariants or of , and 52 of elliptic type 12 Kleiman-Mori cone of a surface 23 nef cone of a surface X 23 set of all exceptional curves (or their classes) on a surface X. the set of exceptional classes of 33 the subset of exceptional classes of 33 the subset of exceptional classes of 33 the subset of exceptional classes of 33 the subset of exceptional classes of 33 the subset of exceptional classes of 34 with square-2 34 the subset of exceptional classes of with square-4 $Y$ the subset of exceptional curves of 33 $Y$ 33 the subset of exceptional curves of $Y$ 33 the subset of exceptional curves of 33 the subset of exceptional curves of all orthogonal primitive roots (they are all $(-2)$ roots) to 36 all $(-2)$ -roots orthogonal to 36 all $(-2)$ -roots orthogonal to 37 all $(-4)$ -roots orthogonal to 36 37 all $(-4)$ -roots orthogonal to 118 a finite quadratic form 118 one of elementary finite quadratic forms 118 the discriminant quadratic form of an even lattice $M$ 29 the eigenspaces of an involution on a module $Q$ $a\in \mathbb{Q}$

\cdots

$(X)/\mathbb{R}^{+}\subset \mathcal{L}(X),$

\cdots

$\mathcal{M}^{(2)}$

\cdots

\cdots

\cdots

\cdots

\ldots

\ldots

\ldots

\ldots

.

$\mathcal{L}(S)$

\ldots

$W_{+}^{(2,4)}$

\ldots

\ldots

\ldots

\ldots

$\mathcal{L}(S)$

\ldots

\ldots

$\mathcal{M}_{+}^{(2,4)}Mod_{M}$

\cdots

$\mathcal{L}(S)$

\ldots

$\mathcal{M}^{(2,4)}$

\cdots

\cdots

\cdots

$\mathcal{M}(X)$

\ldots

\ldots

\ldots

\ldots

\cdots

$Mod_{M\subset L_{K3}}$ $Mod_{(r,a,\delta)}$

$Mod_{S}’$

$N$

$(X, \theta)$

$(r, a, \delta)$

\ldots

$NEF(X)$ $P(X)$ $P(X)_{+}$

$P(X)_{+I}$

$P(X)_{+IIa}$ $P(X)_{+IIb}$

$P(X)_{+III}$ $P^{(2)}(X)_{+}$ $P^{(4)}(X)_{+}$

$P(Y)_{I}$

$P(Y)_{IIa}$ $P(Y)_{IIb}$

$P(Y)_{III}$ $P(\mathcal{M}^{(2)})$

$P^{(2)}(\mathcal{M}^{(2,4)})$

$P^{(2)}(\mathcal{M}_{+}^{(2,4)})$

$P^{(4)}(\mathcal{M}^{(2,4)})$

$P^{(4)}(\mathcal{M}_{+}^{(2,4)})$

\cdots

\cdots

\cdots

$ Q\pm$

$(X, \theta)$

\cdots

\cdots

\cdots

$Z$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$(X, \theta)$

\cdots

\cdots

\ldots

$(X, \theta)$

$(X, \theta)$

$(X, \theta)$

$(X, \theta)$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

$(X, \theta)$

\cdots

$(X, \theta)$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$Y$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\mathcal{M}^{(2)}$

$\mathcal{M}^{(2,4)}$

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

$\mathcal{M}_{+}^{(2,4)}$

$\mathcal{M}^{(2,4)}$

$\mathcal{M}_{+}^{(2,4)}$

\cdots

\cdots

\cdots

\cdots

\cdots

$q_{M}$

\cdots

$Z$

$(k, g, \delta)$

$(r, a, \delta)$

\cdots

$q_{\theta}^{(p)}(p^{k})q$

\ldots

$S$

$(X, \theta)$

\cdots

$NE(Z)$

26 24

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\pm 1$

\cdots

\cdots

$\theta$

\cdots

\cdots

LIST OF NOTATIONS

$\dot{x}w$

$r$

$\tilde{r}$

$(r, a, \delta)$

$R(X, \theta)$ $R_{gen}(X, \theta)$ $S$ $S_{X}$

$U^{(2)}(2^{k})$

$u_{-}^{(2)}(2^{k})$

$u_{+}^{(2)}(2^{k})$

one of main invariants: the Picard number $r= del Pezzo surface the Picard number of a $\log$

$(X, \theta)$

$(X, \theta)$

$V^{(2)}(2^{k})$

\cdots

\cdots

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ..

$(X, \theta)$

\ldots \ldots \ldots \ldots \ldots \ldots \ldots

$Z,$ $Y,$ $(X, \theta)$

one of elementary (2-dimensional) 2-adic lattices one of elementary finite symmetric bilinear forms one of elementary finite quadratic forms \cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\cdots

$V(M)$

\cdots

$V^{+}(X)$

\cdots

\cdots

$W^{(2)}(M)$

$W^{(4)}(M)$ $W^{(2,4)}(M)$

\cdots

$\cdots$

\cdots

\ldots

\ldots

\ldots

\ldots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\cdots$

\cdots

\cdots

\cdots

\cdots

\ldots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

..

$\cdots$

\cdots

\cdots

\cdots

\cdots

$W_{+}^{(2,4)}$

\cdots

the varying part of one of elementary (2-dimensional) 2-adic lattices one of elementary finite symmetric bilinear forms one of elementary finite quadratic forms the light cone $V(M)=\{x\in M\otimes \mathbb{R}|x^{2}>0\}$ of a hyperbolic lattice $M$ the half containing polarization of the light cone $V(S_{X})$ of a surface X the group generated by reflections in all $f\in M$ with $f^{2}=-2$ the group generated by reflections in all $(-4)$ roots of $M$ the group generated by reflections in all $(-2)$ and $(-4)$ roots of $M$ the subgroup of $W^{(2,4)}(S)$ generated by reflections in $(\cdot)$

\cdots

$v_{+}^{(2)}(2^{k})$

\cdots

\cdots

\cdots

$v_{-}^{(2)}(2^{k})$

$rk$ $S$

$Z$

main invariants of the root invariant of the generalized root invariant of $S=(S_{X})_{+}=H^{2}(X, \mathbb{Z})_{+}$ , the main invariant of the Picard lattice (modulo torsion) of a surface X $Z,$ $Y,$

\cdots

$Var(\cdot)$

rk S_{Y};r=$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots

see Proposition 2.2 see (50) the Weyl group of a finite root system \cdots

$W_{+}^{(4)}(\mathcal{M}^{(2)})$

\cdots

$W(R)$ $X$ $(X, \theta)$ $X^{\theta}$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\cdots$

$(Y, C)$ $Z$ $\alpha$

\cdots

\cdots

$R$

\cdots

\cdots

\cdots

\cdots \cdots \cdots \cdots \cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$\cdots$

\cdots

\cdots

\cdots

\ldots \ldots \ldots \ldots

$\theta$

\cdots

\cdots

\cdots

\ldots

\cdots

\ldots

\ldots

\cdots

\cdots

\ldots

\ldots

..

$\leq 2$

$\log$

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

..

$0$

23 23 36 36

\ldots

..

\ldots

\ldots

35 36 37 2 22 22 31 20 20 18 41

$\delta_{S}=0$

$(X, \theta)$

\ldots

$0$

\ldots

$0$

$H_{2}(X, \mathbb{Z})$

$S$

$\delta_{M}$

$\cdots$

$\theta$

\ldots

$\delta$

\cdots

K3 surface X K3 surface $X$ with a non-symplectic involution fixed points of involution on X DPN surface; also $Y=X/\{1, \theta\}$ , also right resolution of $Z$ .. a DPN pair; also $(Y=X/\{1, \theta\}, C=X^{\theta})$ del Pezzo surface $Z$ of index the invariant ( or 1) of the root invariant $R(X, \theta)$ ; zero iff 0 or 1; zero Oiff $X^{\theta}\sim 0mod 2$ in for the main invariant of $Z,$ $Y$ or or 1; one of invariants of a 2-elementary lattice $M$ , zero iff for any $m^{*}\in M^{*}$ \cdots

$Y$

26

$\triangle_{+}^{(2,4)}\subset$

$\Delta^{(2,4)}(S)$ $W^{(4)}(\mathcal{M}^{(2)})$

29 106 30 40 42 29 23 118 119 118 62 118 119 118

\ldots

\ldots

\ldots

30

$0$

$(m^{*})^{2}\in \mathbb{Z}$

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots

30

XV

LIST OF NOTATIONS $\triangle_{+}^{(2)}$

$\triangle_{+t}^{(2)}$

the set of all $(-2)$ -roots of $S=(S_{X})_{+}$ the subset of $(-2)$ -roots of $S=(S_{X})_{+}$ the subsets of $(-4)$ -roots of $\triangle^{(2)}(S)$

\ldots

$\triangle_{\pm}^{(4)}$

$\triangle_{+}^{(2,4)}$

$\triangle_{-}^{(6)}$

$(S_{X})_{\pm}$

it is

$\triangle(4)(S)$

$\triangle(4)(\mathcal{M}^{(2)})$

$\Gamma(\cdot)$

the subset of $(-6)$ -elements of the set of all $(-2)$ -roots of a lattice the set of all $(-4)$ -roots of a lattice see Proposition 2.2 see(51)

\cdots

\cdots

\cdots

\ldots

.

\ldots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

.

$S$ \cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

$S$

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

; ; equivalent to Gram matrix of the Dynkin diagram of the dual graph of exceptional curves of period domain of K3 surfaces with condition $M\subset L_{K3}$ on Picard lattice period domain of K3 surfaces with condition $M\subset L_{K3}$ on Picard lattice with forgetting a part (the homomorphism) of the root invariant $R(X, \theta)$ the discriminant group $\mathfrak{A}_{M}=M^{*}/M$ of a lattice $M$ the orthogonal sum of lattices (with very few exceptions when it is used to denote the direct sum of modules) $(.$

\cdots

$\Omega_{M\subset L_{K3}}$

\ldots

\ldots

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\ldots

\cdots

\ldots

\cdots

\ldots

\ldots

\ldots

\cdots

\cdots

\cdots

..

$\mathcal{M}$

\ldots

$\xi$

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

\ldots

$\mathfrak{A}_{M}$

\cdots

$\oplus$

34 34 34 35 34 34 36 36 37

$(\cdot)$

$)$

$(\cdot)$

$\tilde{\Omega}_{M\subset L_{K3}}$

\ldots

\ldots

\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots

\cdots

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

\ldots

\ldots

$\triangle_{+}^{(2)}\cup\triangle_{+}^{(4)}$

$(S_{X})_{-}$

$\triangle(2)(S)$

\ldots

\ldots

\cdots

\cdots

..

\cdots

\cdots

\cdots

\cdots

50

27 27 40 45

117

List of Tables

1

Fundamental chambers of reflection groups $W^{(2,4)}(S)$ for 2-elementary even hyperbolic lattices of elliptic type of Diagrams $\Gamma(P(X)_{+})$ of extremal K3 surfaces elliptic type which are different from Table 1 Dual diagrams of all exceptional curves of extremal .ight DPN surfaces of elliptic type Correspondence between connected components of and singularities of $\mathcal{M}^{(2,4)}$

$S$

\cdots

2

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

64

$\Gamma$

\cdots

4

52

$(X, \theta)$

\cdots

3

\cdots

\cdots

\cdots

\cdots

\cdots

\cdots

93

$\Gamma$

$D=\sigma(\tilde{D})$

\cdots

xi

\cdots

\cdots

\cdots

103

Introduction

The main purpose of this work is to classify del Pezzo surfaces with logterminal singularities of index one or two. By classification, we understand a description of the intersection graph of all exceptional curves on an appropriate (called right) resolution of singularities together with the subgraph of the curves which are contracted to singular points. The final results are similar to classical results about classification of non-singular del Pezzo surfaces and use the usual finite root systems. However, the intermediate considerations use K3 surfaces and reflection groups

in hyperbolic spaces. The work is self-contained and can also serve as an introduction to del Pezzo and K3 surfaces. It is based on our paper [AN88]. See also [AN89] for a short exposition of these results. In this work, we consider algebmic varieties over thefield ofcomplex numbers, and do not mention this further. $\mathbb{C}$

0.1. Historical remarks and our main principle of classification oflog del Pezzo surfaces of index

$\leq 2$

A complete algebraic surface $Z$ with terminal singularities is a del Pezzo surface if its anticanonical $divisor-K_{Z}$ is ample. $A$ 2-dimensional terminal singularity over is a singularity which is analytically equiv, where alent to a quotient singularity is a finite subgroup. The index of $z\in Z$ is the minimal positive integer for which the divisor is a Cartier divisor in a neighbourhood of The aim of this work is to classify del Pezzo surfaces with terminal singularities (or simply del Pezzo surfaces) of index include classical cases of nondel Pezzo surfaces of index singular del Pezzo surfaces and del Pezzo surfaces of index 1, i.e. $\log$

$\log$

$\mathbb{C}$

$ G\subset$

$\mathbb{C}^{2}/G$

$GL(2, \mathbb{C})$

$i$

$iK_{Z}$

$z.$

$\log$

$\log$

$\leq 2.$

$\leq 2$

${\rm Log}$

$\log$

1

INTRODUCTION

2

del Pezzo surfaces. Let us recall some classical results Gorenstein about these del Pezzo surfaces. In 1849, Cayley [Ca1849] and Salmon [Sa1849] discovered 27lines on a non-singular cubic surface . Now we know that they are all exceptional curves on a non-singular del Pezzo surface of degree 3, and that they are crucial for its geometry. Here, the degree of a del Pezzo surface is $\log$

$Z$

$Z$

$Z$

$d$

$d=(K_{Z})^{2}.$

Classification of nonsingular del Pezzo surfaces is well known, and they are classical examples of rational surfaces (see, e.g. [Nag60, Man86, $MT$ 86] . $A$ connection between nonsingular del Pezzo surfaces and reflection groups was noticed a long time ago. Schoutte [Sch10] noted that there is an incidence-preserving bijection between 27 lines on a smooth cubic and vertices of a certain polytope in . In modem terminology, this polytope is the convex hull of an orbit of reflection group $W(E_{6})$ . Coxeter [Cox28] and Du Val [DV33] noted a similar correspondence between $(-1)$ -curves on del Pezzo surfaces of degree 2 and 1 and reflection polytopes for groups $W(E_{7})$ and $W(E_{8})$ . Du Val was the first to investigate the relationship between reflection groups and singular surfaces. In $[DV34a]$ he introduced Du Val singularwere classified by ities. Possible singularities of cubic surfaces Schl\"afli [Sc1863] and Cayley [Ca1869]. In $[DV34b]$ Du Val found all possible configurations of Du Val singularities on the “surfaces of del Pezzo ramified in a quartic series” of degree 2 and 1, i.e. double covers over a quadratic cone ramified in an intersection and double covers $Q$ of with a cubic. As was proved much later [Dem80, HW81], these are del Pezzo surfaces of degree 2 and 1. precisely the Gorenstein Du Val observed the following amazing fact: the configurations of sinof the degree with Du Val singugularities on del Pezzo surfaccs larities are in a one-to-one correspondence with subgroups generated by , i.e. reflections (i.e. root subsystems) of a reflection group of type $A_{2}+A_{1}$ respectively for $d=1,$ , with four excep$8A_{1},7A_{1},$ for $d=2$ . (These days, we for $d=1$ and tions: know that the prohibited cases do appear in characteristic 2.) He also noted ) there are two non-conjugate in that in some cases (for example and, on the other hand, there are two distinct ways to embed a subgroup deformation types of surfaces. The proof was by comparing two long lists. The reflection subgroups were conveniently classified by Coxeter [Cox34] in the same 1934 volume of Proceedings of Cambridge Philosophical Society. Du Val went through and sextic curves on the quadratic cone all possibilities for quartics on $Q$ and computed the singularities of the corresponding double covers $)$

$\mathbb{R}^{6}$

$Z_{3}\subset \mathbb{P}^{3}$

$Z_{2}\rightarrow \mathbb{P}^{2}$

$Z_{1}\rightarrow Q$

$\log$

$d$

$Z_{d}$

$E_{9-d}$

$6$

$E_{8},$ $E_{7},$ $E_{6},$ $D_{5},$ $A_{4},$

$\ldots,$

$7A_{1}$

$D_{4}4A_{1}$

$4A_{1}$

$E_{8}$

$\mathbb{P}^{2}$

$Z_{d},$

0.1. HISTORICAL REMARKS AND MAIN PRINCIPLE OF CLASSIFICATION

3

$d=1,2$ . The modem explanation for the fact that configurations of singularities correspond to some reflection subgroups is simple: $(-2)$ -curves

on the minimal resolution of a Gorenstein del Pezzo which is a root lattice of type $Y$

$(K_{Y})^{\perp}$

$Z$

lie in the lattice

$E_{9-d}.$

In the $1970s$ , Gorenstein del Pezzo surfaces attracted new attention in connection with deformations of elliptic singularities, see [Loo77, Pin77, $BW$ 79]. The list of possible singularities was rediscovered and reproved using modem methods, see [HW81, Ura83, BBD84, Fur86]. In addition, Demazure [Dem80], and Hidaka and Watanabe [HW81] established a fact which Du Val intuitively understood but did not prove, lacking modem definitions and tools: the minimal resolutions of Gorenstein del Pezzo surfaces are precisely the blowups of $9-d$ points on in “almost general position”, and is obtained from such blowup by contracting all $(-2)$ -curves. $Y_{d}$

$\log$

$Z_{d}\neq \mathbb{P}^{1}\times \mathbb{P}^{1}$

$\mathbb{P}^{2}$

$Z_{d}$

In addition to clarifying, unifying and providing new results for the index 1 case, our methods are general enough to obtain similar results in the much more general case of log del Pezzo surfaces $Z$ of index . Thus, we admit $\log$ terminal singularities of index 1 and index 2 as well. Classification of the much larger class of $\log$ del Pezzo surfaces of index (together with the described above classical index-l case) is the subject of our work. By classification, we understand a description of the dual graphs of all exceptional curves ( . irreducible with negative self-intersection) on an , together with the subset appropriate resolution ofsingularities : of curves contmcted by . We call the right resolution. See Section 0.3 below for the precise definition. For Gorenstein, i. e. of index 1 singularities, is simply the minimal resolution. Thus, in the principle of classification we follow the classical discovery by Cayley [Ca1849] and Salmon [Sa1849] of 27lines on a non-singular cubic surface which we have mentioned above. The dual graph of exceptional curves provides complete information about the surface. Indeed, knowing the dual graph of exceptional curves on $Y$ , we can describe all the ways to obtain $Y$ and $Z$ by blowing up $n=0,2,3\ldots.$ from the relatively minimal rational surfaces or Images of exceptional curves on $Y$ then give a configuration of curves on related with these blow ups. Vice versa, if one starts with a similar” configuration of curves on and performs“similar” blowups then the resulting surface $Y$ is guaranteed to be the right resolution of a $\log$ del Pezzo surface $Z$ , by Theorem 3.20. of index $\leq 2$

$\leq 2$

$i.$

$e$

$\sigma$

$\sigma$

$Y\rightarrow Z$

$\sigma$

$\sigma$

$Y\rightarrow\overline{Y}$

$\overline{Y}=\mathbb{P}^{2}$

$\mathbb{F}_{n},$

$\overline{Y}$

$\overline{Y}$

$\leq 2$

4

INTRODUCTION

In the singular case of index 1, we add to the classical results which were described above a description of all graphs of exceptional curves on this was the minimal resolution of singularities. In the case of Pic done by Bindschadler, Brenton and Dmcker [BBD84]. $Z=\mathbb{Z}$

0.2. Classification of log del Pezzo surfaces of index and K3 surfaces

$\leq 2$

del Pezzo surfaces The main method for obtaining our classification of of index is to reduce it to a classification of K3 surfaces with nonsymplectic involution and to K3 surfaces theory. The main points of the latter are contained in [ $Nik80a$ , Nik79, Nik83, $Nik84a$ , Nik87]. del Pezzo surface $Z$ of index In Chapter 1, we show that on each the linear system $|-2K_{Z}|$ contains a nonsingular curve, and that there exists for which the an appropliate (right”) resolution of singularities : $C$ linear system $|-2K_{Y}|$ contains a nonsingular divisor (i.e. $Y$ is a right DPN surface) such that the component of $C$ that belongs to $\sigma^{*}|-2K_{Z}|$ has (i.e. the DPN surface $Y$ is of elliptic type). genus In Chapter 2, following [ $Nik80a$ , Nik79, Nik83, $Nik84a$ , Nik87], we build a general theory of DPN surfaces $Y$ . Here, we use the fact that the double cover $X$ of $Y$ branched along $C$ is a K3 surface with a nonsymplectic involution . In this way, the classification of DPN surfaces $Y$ and DPN pairs $(Y, C)$ is equivalent to the classification of K3 surfaces . The switch to K3 surfaces is imporwith non-symplectic involution tant because it is easy to describe exceptional curves on them and there are powerful tools available: the global Torelli Theorem [PS-Sh71] due to Piatetsky-Shapiro and Shafarevich, and surjectivity of the period map [Ku177] due to Vik. Kulikov. In Chapter 3, we extend this theory to the classification of DPN surfaces $Y$ , by of elliptic type, i.e. when one of the components of $C$ has genus describing dual diagrams of exceptional curves on $Y$ . See Theorems 3.18, 3.19 and 3.20. In Section 3.6, we give an application of this classification (and $D\in|-2K_{F_{n}}|$ as to a classification of curves $D$ of degree 6 on well) with simple singulanities, in the case when one of components of $D$ has geometric genus In obtaining results of Chapters 2 and 3, a big role is played by the arithmetic of quadratic forms and by reflection groups in hyperbolic spaces which are very important in the theory of K3 surfaces. From this point of view, the success of our classification hinges mainly on the fact that we explicitly describe some hyperbolic quadratic forms and their subgroups generated by all reflections (2-elementary even hyperbolic lattices of small $\log$

$\leq 2$

$\leq 2$

$\log$

$\sigma$

$Y\rightarrow Z$

$\geq 2$

$\theta$

$(X, \theta)$

$\geq 2$

$\mathbb{P}^{2}$

$\geq 2.$

5

0.3. FINAL CLASSIFICATION RESULTS

computations are also important by themselves for the arithmetic of quadratic forms. In Chapter4, the results of Chapters 1–3 are applied to the classifica. In particular, we show that tion of del Pezzo surfaces of index there are exactly $ 18\log$ del Pezzo surfaces of index 2 with Picard number 1. For completeness, we also included the list of the isomorphism classes in the index 1 Picard number 1 case. This list, which for the most difficult degree 1 case can be deduced from [MP86], is skipped or given with some inaccuracies in other references. In Section 4.3, following [BBD84], we give an application of our classification to describe some rational compactifications of certain affine surfaces. In Section 4.4, we give formulae for the dimension of moduli spaces of log del Pezzo surfaces of index In Section 2.2 we review results about K3 surfaces over which we use. In Appendix, for reader’s convenience, we review known results about lattices, discriminant forms of lattices, non-symplectic involutions on K3 $A$ .3). For instance, in Section A.2 we which we use (see Sections A. (see below) of nonreview the classification of main invariants symplectic involutions on $K3$ and their geometric interpretation, which are very important in this work. In Section A.4 we give details of calculations of fundamental chambers of hyperbolic reflection groups which were skipped in the main part of the work. They are very important by themselves. Thus, except for some standard results from Algebraic Geometry (mainly about algebraic surfaces), and reflection groups and root systems, our work is more or less self-contained. rank,

see Theorem

3.1). These

$\leq 2$

$\log$

$\leq 2.$

$\mathbb{C}$

$I$

–

$(r, a, \delta)$

0.3. Final classification results for index

$\log$

del Pezzo surfaces of

$\leq 2$

we try to give an explicit and as elementary exposition as possible of our final results on classification of log del Pezzo surfaces of index

Below,

$\leq 2.$

In spite of importance of K3 surfaces, in the fina] classification results K3 surfaces disappear, and it is possible to formulate all results in tenns of only del Pezzo surfaces and their appropriate non-singular models, which are DPN surfaces. del Pezzo surface of index $\leq 2$ . Its singularities of Let $Z$ be a index 1 are Du Val singularities classified by their minimal resolution of or , with singularities. They are described by Dynkin diagrams $Z$ each vertex having weight-2. Singularities of of index 2 are singularities which have minimal resolutions with dual graphs shown below: $\log$

$A_{n},$

$K_{n}$

$D_{n}$

$E_{n}$

INTRODUCTION

6

, additionally one has To get the right resolution of singularities : to blow up all points of intersection of components in preimages of singular is described by . Then the right resolution of a singular point points the graph $Y\rightarrow Z$

$\sigma$

$K_{n}$

$K_{n}$

2n-l vertices

In these graphs every vertex corresponds to an irreducible non-singular rational curve with equal to the weight of the vertex. Two vertices are connected by an edge if $F_{i}\cdot F_{j}=1$ , and are not connected if $F_{i}\cdot F_{j}=0.$ Thus, the right resolution of singularities : del Pezzo of a is obtained by taking the minimal resolutions of the surface of index singular points of index 1 and the right resolutions, as in the above figure, of index 2. of the singular points and the Our classification of $logdel$ Pezzo surfaces $Z$ of index corresponding $DPN$ surfaces of elliptic type which are right resolutions of singularities of $Z(i.$ . thy are appropriate non-singular models of the $del$ Pezzo surfaces) is contained in Table 3 (see Section 3.5). All cases of Table 3 are labelled by a number $1\leq N\leq 50$ . For $N=$ $7,8,9,10,20$ we add some letters and get cases: 7a,b, 8arc, $9a-f,$ $10a-m,$ $20a-d$ . Thus, altogether, Table 3 contains $F_{i}$

$F_{i}^{2}$

$Y\rightarrow Z$

$\sigma$

$\log$

$\leq 2$

$K_{n}$

$\leq 2$

$e$

$50+(2-1)+(3-1)+(6-1)+(13-1)+(4-1)=73$ cases. $50$ enumerate the so-called main invariants The labels $N=1,$ (equivalently $(k=$ of del Pezzo surfaces . They are triplets $(r-a)/2,$ $g=(22-r-a)/2,$ where are integers: $r\geq 1,$ $a\geq 0,$ $\ldots,$

$\log$

$Z$

$(r, a, \delta)$

$\delta))$

$\delta$

$r,$

$a,$

7

0.3. FINAL CLASSIFICATION RESULTS $\delta\in\{0,1\},$ $g\geq 2,$ $k\geq 0$

.

Thus, there exist exactly 50 possibilities for the

) of log del Pezzo surfaces of main invariants index The main invariants have a very important geometric meaning. Any and its right resolution of singularities $Y$ del Pezzo surface of index are rational. The number $r=$ rk Pic $Y$ . We prove that $|-2K_{Z}|$ is the Picard number of $Y$ , i. e. Pic of genus $g\geq 2$ which explains contains a non-singular irreducible curve the geometric meaning of . This is equivalent to saying that there is a curve $(r, a, \delta)$

$(equivalently, (k, g, \delta)$

$\leq 2.$

$\log$

$Z$

$\leq 2$

$Y=\mathbb{Z}^{r}$

$C_{g}$

$g$

$C=C_{g}+E_{1}+\cdots+E_{k}\in|-2K_{Y}|,$

are all exceptional curves on $Y$ with $(E_{i})^{2}=-4$ . (The inequality $g\geq 2$ means that $Y$ is of elliptic type). All these curves come from the where

$E_{i}$

$E_{i}$

right resolution of singularities of described above. Thus, the invariant equals the number of exceptional curves on $Y$ with square-4. All of them are nonsingular and rational. .g., $k=0$ if and only if $Z$ is Gorenstein and all of its singularities are Du Val, see Chapter 1. Let us describe the invariant $\delta\in\{0,1\}$ . The components are disjoint. Since $C$ is divisible by 2 in Pic $Y$ , it defines a double cover : ramified in $C$ . Let be the involution ofthe double cover. Then the set of fixed points $X^{\theta}=C$ . Here, $X$ is a K3 surface and $Z$

$k$

$E$

$C_{g},$

$E_{1},$

$\ldots,$

$E_{k}$

$\pi$

$X\rightarrow Y$

(1)

$\theta$

$\delta=0\Leftrightarrow X^{\theta}\sim 0$

there exist signs (2)

$(\pm)_{i}$

$mod 2$

in

$H_{2}(X, \mathbb{Z})$

$\Leftrightarrow$

for which $\frac{1}{4}\sum_{i}(\pm)_{i}cl(C^{(i)})\in$

Pic

$Y,$

where are all irreducible components $(i. e. C_{g}, E_{1}, \ldots, E_{k})$ of $C.$ As promised, our classification describes all intersection (or dual) graphs of exceptional curves on $Y$ and also shows exceptional curves which del Pezzo surface $Z$ of : to get the must be contracted by index from $Y$ . All of these graphs can be obtained from graphs in . Let us the right column of Table 3 of the same main invariants describe this in more details. All exceptional curves $E$ on $Y$ are irreducible, non-singular, and rational. They are of three types: of $ C\in$ (1) $E^{2}=-4$ , equivalently $E$ is a component of genus $|-2K_{Y}|$ . In the graphs of Table 3 these correspond to double transparent vertices; (2) $E^{2}=-2$ . In the graphs of Table 3 these correspond to black vertices; $C^{(i)}$

$\Gamma(Y)$

$\sigma$

$Y\rightarrow Z$

$\log$

$\Gamma$

$\leq 2$

$(r, a, \delta)$

$0$

8

INTRODUCTION

(3)

$E^{2}=-1$

(the lst kind). In the graphs of Table

3 these correspond

to simple transparent vertices. , together with with $(E_{i})^{2}=-4,$ $i=1,$ All exceptional curves all exceptional curves $F$ of the lst kind such that: there exist two different $i\neq j$ , with $(E_{i})^{2}=(E_{j})^{2}=-4$ and $F\cdot E_{i}=F\cdot E_{j}=1,$ curves , the logarithmic part of surface $Y$ . Since define $F\cdot(-2K_{Y})=F\cdot C=2$ , the curves $F$ are characterized by the property $C_{g}\cdot F=0$ . The logarithmic part can be easily seen on graphs of $i=1,$ , are shown as double transparent Table 3: curves are shown as simple transparent the curves $F$ of the first kind of vertices connected by two edges with (always two) double transparent verand is also called the logarithmic tices. This part of is denoted by part of graph Thus, we have: $\ldots k$

$E_{i}$

$E_{i},$

$E_{j},$

${\rm Log}\Gamma(Y)\subset\Gamma(Y)$

${\rm Log}\Gamma(Y)$

$\Gamma$

$k$

$E_{i},$

$ve\iota tices,$

$\ldots,$

${\rm Log}\Gamma(Y)$

$\Gamma$

$\Gamma$

${\rm Log}\Gamma$

.

(3)

${\rm Log}(\Gamma(Y))={\rm Log}\Gamma$

The logarithmic part gives precisely the preimage ofsingular points of of index two. All exceptional curves $E$ on $Y$ with $E^{2}=-2$ define Duv , the Du Val part of surface $Y$ Its connected components are of or and they correspond to all Du Val singularities Dynkin graphs gives precisely the preimage of $Z$ . Thus the $DuVal$ part Duv of all $DuVal$ ( . of index one) singular points of $Z$ . For each of the graphs of Table 3, the Du Val part of graph , is defined by all of its black vertices. We have: $(with the same main invariants (r, a, \delta)$ ).

${\rm Log}\Gamma(Y)$

$Z$

$\Gamma(Y)\subset\Gamma(Y)$

.

$\Gamma(Y)$

$A_{n},$ $D_{n}$

$E_{n}$

$\Gamma(Y)\subset\Gamma(Y)$

$i.$

$e$

$\Gamma$

(4)

$D=$

Duv

$\Gamma(Y)\subset$

$(for the same main invariants (r, a, \delta)$ ).

Duv

$\Gamma$

Any subgraph

taken. Let us describe the remaining part of defines a lattice in the usual way. It is

$\Gamma(Y)$

$D$

of Duv can be

. Each graph

$\Gamma$

$\Gamma$

of Table 3

$S_{Y}$

$S_{Y}=(\bigoplus_{v\in V(\Gamma)}\mathbb{Z}e_{v})/Ker$

defined by the intersection pairing: $e_{v}^{2}=-1$ , if is simple transparent, $e_{v}^{2}=-2$ , if is black, $e_{v}^{2}=-4$ , if is double transparent, $e_{v}\cdot e_{v’}=m$ $v\neq v’$ are connected by $m$ edges. Here, means the direct if the vertices $Ker$ denotes the kemel of this pairing. We denote sum of -modules, and $E_{v}=e_{v}$ mod $Ker$ . In all cases except the trivial cases $N=1$ when $N=3$ when , and $N=11$ $N=2$ when or $Y.$ , the lattice gives the Picard lattice of when $v$

$v$

$v$

$\oplus$

”

$\mathbb{Z}$

$Y=\mathbb{P}^{2},$

$Y=\mathbb{F}_{4}$

$Y=\mathbb{F}_{0}$

$S_{Y}$

$\mathbb{F}_{2},$

$Y=\mathbb{F}_{1}$

9

0.3. FINAL CLASSIFICATION RESULTS

Duv define divisor and $D=$ Duv $ v\in V({\rm Log}\Gamma(Y)\cup$ Duv , of the corresponding exceptional classes curves on $Y$ . Each exceptional curve $E$ is evidently defined by its divisor class. Black vertices $v\in V$ (Duv ) define roots $E_{v}\in S_{Y}$ with $E_{v}^{2}=-2$ and such define reflections in these roots which are automorphisms of that $s_{E_{v}}(E_{v})=-E_{v}$ and gives identity on the orthogonal complement $v\in V$ (Duv ), generate a finite . These reflections in to Weyl group $W\subset O(S_{Y})$ . The remaining part Thus,

${\rm Log}\Gamma(Y)={\rm Log}\Gamma$

$\Gamma$

$\Gamma(Y)\subset$

$\Gamma(Y))$

$E_{v},$

$\Gamma$

$S_{Y}$

$s_{E_{v}}$

$s_{E_{v}}$

$\Gamma$

$S_{Y}$

$E_{v}$

$E_{v}^{\perp}$

$s_{E_{v}},$

Var $\Gamma(Y)=\Gamma(Y)-({\rm Log}\Gamma(Y)\cup$ Duv (it

is called the varying part of surface

$Y$ )

$\Gamma(Y))$

is defined by

Var $\Gamma=\Gamma-(Duv\Gamma$ ULog

$\Gamma)$

of the graph of Table 3. Further, we identify exceptional curves $ v\in$ $V(\Gamma(Y))$ with their divisor classes $E_{v}\in S_{Y}$ . We have the main formula $\Gamma$

(5)

$V$

(Var

$\Gamma(Y)$

)

$=\{E\in W(\{E_{v}|v\in V(Var \Gamma)\})|E\cdot D\geq 0\}\subset S_{Y}$

which describes Var completely. Here $E\cdot D\geq 0$ means $E\cdot E_{i}\geq 0$ then defines the full graph for any $E_{i}\in D$ . The intersection pairing on $Y$ . of This completes the description of possible graphs of $Z$ exceptional curves of log del Pezzo surfaces of index Thus, to find all possible graphs of exceptional curves of : $Z$ , one has to choose one of the graphs of Table 3 (this also defines main of $Y$ and $Z$ ), then choose a subgraph $D=$ Duv invariants Duv . Then consists of $D,$ ${\rm Log}\Gamma(Y)={\rm Log}\Gamma$ and the remaining defined by the formula (5), the elements in the $W$ -orbits of part Var Var that have non-negative intersection with the Du Val part. See Theorems 3.18, 3.19, 3.20 and 4.1. See Section 4.2 about such type of calculations in the most non-trivial case $N=20.$ We note two important opposite cases. Extremal case. This is the case when $D=$ Duv $\Gamma(Y)=$ Duv . Then $\Gamma(Y)=\Gamma$ is completely calculated in Table 3. This case is called extremal and gives del Pezzo surfaces $Z$ with Du Val singularities of the highest ) is rank, respectively rk Pic $Z=r-\# V({\rm Log}\Gamma(Y))-\# V$ (Duv minimal for the fixed main invariants. In particular, this case delivers all the with rk Pic $Z=1.$ del Pezzo surfaces of index cases of minimal See Theorems 3.18, 4.2, 4.3. No $DuVal$ singularities. This is the case when $D=$ Duv Equivalently, all singularities of $Z$ have index 2, if they exist. Then $\Gamma(Y)=$ $\Gamma(Y)$

$S_{Y}$

$\Gamma(Y)$

$\Gamma(Y)$

$\leq 2.$

$\Gamma(Y)$

$\sigma$

$ Y\rightarrow$

$\Gamma$

$\Gamma(Y)\subset$

$(r, a, \delta)$

$\Gamma$

$\Gamma(Y)$

$\Gamma(Y)$

$\Gamma$

$\Gamma$

$\log$

$\Gamma(Y)$

$\log$

$\leq 2$

$\Gamma(Y)=\emptyset.$

10

INTRODUCTION

${\rm Log}\Gamma\cup$

Var

where

$\Gamma(Y)$

(6)

$V$

(Var

$\Gamma(Y)$

)

$=W(\{E_{v}|v\in V($ Var

$\Gamma)\})$

.

ac, $9a-f,$ $10a-m,$ $20a-d$ give the same graphs (because they have the same, equal to zero, root invariant, see below), and one can always take the cases $7a,$ $8a,$ $9a,$ $10a,$ $20a$ for the main invariants. This case is very similar to and includes the classical case ofnonsingular del Pezzo surfaces corresponding to the cases 1–10. See Theorem 4.4 about this (without Du Val singularities) case. del Pezzo surfaces up to deformation. of this case are defined by their main invariants The Du Val parts Duv of graphs of Table 3 in this case can be considered to be analogs of root systems (or Dynkin diagrams) which one usually associates to non-singular del Pezzo surfaces. Its true meaning is to give the type of the Weyl group $W$ that describes the valying part Var $(\Gamma(Y))$ from Var by the formula (6). In the cases 7–10, 20, one can take graphs of the cases $7a-10a,$ $20a.$ The Root invariant. It is possible that two different subgraphs Duv Duv of graphs of Table 3 (with the same main invariand give isomorphic graphs ants for the corresponding right resolutions, and then they give similar $\log$ del Pezzo surfaces $Z$ and $Z’$ , according to our classification. The root invariant of index Here, all the multiple

cases

$7a,b,$

$8$

${\rm Log}$

$(r, a, \delta)$

$\Gamma$

$\Gamma$

$\Gamma$

$\Gamma$

$ D\subset$

$\Gamma,$

$ D\subset$

$\Gamma’$

$(r, a, \delta))$

$\Gamma(Y)$

$\Gamma(Y’)$

$\leq 2$

(7)

$([D], \xi)$

gives the necessary and sufficient condition for this to happen. To define the root invariant (7), we first remark that the main invariants define a unique hyperbolic ( . with one positive square) even 2elementary lattice with these invariants. Here $r=$ rk for any $x\in S^{*}$ In (7), $[D]$ is the and $\delta=0$ , if and only if root lattice generated by $D$ , and : $[D]/2[D]\rightarrow S^{*}/S$ a homomorphism $i.$

$(r, a, \delta)$

$e$

$S$

$S,$ $S^{*}/S\cong(\mathbb{Z}/2)^{a},$

$(x^{*})^{2}\in \mathbb{Z}$

$\xi$

preserving finite forms $(x^{2})/2mod 2,$ $x\in[D]$ , and $y^{2}mod 2,$ $y\in S^{*}.$ The construction of the root invariant (7) uses the double cover : $Y$ by a K3 surface $X$ (see above) with the non-symplectic involution Then is the sublattice where acts as identity. The root invariant (7) is considered up to automorphisms of and the root lattice $[D].$ See Sections 2.5 and 3.2 about this construction and a very easy criterion (the kemel $H$ of is almost equivalent to ) for two root invariants to be isomorphic. The root invariant was first introduced and used in $[Nik84a]$ and [Nik87]. In practise, to calculate the root invariant of a $\log$ del Pezzo surface of , one shouldjust go from the graphs of Table 3 to the equivalent index or $\Gamma(P(X)_{+})$ of Tables 1 or 2 of exceptional curves graphs (see Sections 3.2, 3.5). for the K3 pairs $\pi$

$ X\rightarrow$

$\theta.$

$S=H^{2}(X, \mathbb{Z})^{\theta}$

$\theta^{*}$

$S$

$\xi$

$\xi$

$\Gamma$

$\leq 2$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$(X, \theta)$

11

0.3. FINAL CLASSIFICATION RESULTS

Duv Duv of graphs of Table 3 give isomorphic full graphs and of their del Pezzo surfaces if and only if their root invariants (7) are isomorphic (see Theorem 3.5). Moreover, we constantly use the root invariant to prove existence of the corresponding K3 pairs and del Pezzo surfaces . The main invariants and the root invariants (7) are the main tools in our classification. They are equivalent to the full graphs of exceptional $Y$ curves on , but they are much more convenient to work with. For nonsingular del Pezzo surfaces and del Pezzo surfaces of index without Du Val singularities the root invariant is zero. This is why, in these cases, we have such a simple classification as above. See Section 4.2 about enumeration of root invariants (equivalently of graphs of exceptional curves) in the most non-trivial case $N=20.$ Thus, two Du Val subgraphs

$ D\subset$

$\Gamma,$

$\Gamma’$

$\log$

$\Gamma(Y’)$

$\Gamma(Y)$

$(X, \theta)$

$ D\subset$

$Z$

$\log$

$(r, a, \delta)$

$\Gamma(Y)$

$\log$

$\leq 2$

Acknowledgments. We are grateful to V.A. Iskovskikh for useful discussions. The first author was supported by NSF for part of this work. The second author is grateful to Steklov Mathematical Institute, Max-PlanckInsitut f\"ur Mathematik and the University of Liverpool for hospitality. We also would like to thank the Referees for many helpful suggestions.

1 Log del Pezzo surfaces of index Divisor Theorem

$\leq 2$

and Smooth

1.1. Basic definitions and notation be a canonical Weil divisor on Let $Z$ be a normal algebraic surface, and it. The surface $Z$ is called -Gorenstein if a certain positive multiple of is Cartier, and -factorial if this is true for any Weil divisor $D$ . These properties are local: one has to require all singulanities to be -Gorenstein, respectively -factorial. Let us denote by $Z^{1}(Z)$ and $Div(Z)$ the groups of Weil and Cartier divisors on $Z$ . Assume that $Z$ is -factorial. Then the groups of -Cartier divisors and -Weil divisors coincide. The and intersection form defines natural pairings $K_{Z}$

$\mathbb{Q}$

$K_{Z}$

$\mathbb{Q}$

$\mathbb{Q}$

$\mathbb{Q}$

$Z^{1}(Z)\otimes \mathbb{Q}$

$\mathbb{Q}$

$Div(Z)\otimes \mathbb{Q}$

$\mathbb{Q}$

$\mathbb{Q}$

$Div(Z)\otimes \mathbb{Q}\times Div(Z)\otimes \mathbb{Q}\rightarrow \mathbb{Q},$

$Div(Z)\otimes \mathbb{R}\times Div(Z)\otimes \mathbb{R}\rightarrow \mathbb{R}.$

Quotient groups modulo kemels of these pairings are denoted and $Z$ projective, is are they finite-dimensiorespectively; if the surface nal linear spaces. The Kleiman-Mori cone is a convex cone $NE(Z)$ in , the closure of the cone generated by the classes of effective curves. $N_{\mathbb{Q}}(Z)$

$N_{\mathbb{R}}(Z)$

$N_{\mathbb{R}}(Z)$

-Cartier divisor on . We will say that $D$ is ample if some positive multiple is an ample Cartier divisor in the usual sense. By Kleiman’s criterion [Kle66], for this to hold it is necessary and sufficient that $D$ defines a strictly positive linear function on $NE(Z)-\{O\}.$ terminal singularities if One says that the surface $Z$ has only it is -Gorenstein and for one (and then any) resolution of singularities , in a natural fonnula $K_{Y}=\pi^{*}K_{Z}+\sum\alpha_{i}F_{i}$ , where are : $\alpha_{i}>-1$ , . The least common one has irreducible divisors and multiple of denominators of is called the index of Let

$D$

be a

$Z$

$\mathbb{Q}$

$\log$

$\mathbb{Q}$

$\pi$

$Y\rightarrow Z$

$F_{i}$

$\alpha_{i}\in \mathbb{Q}$

$Z.$

$\alpha_{i}$

12

1.2. LOG TERMINAL SINGULARITIES OF INDEX 2

13

It is known that two-dimensional terminal singularities in characteristic zero are exactly the quotient singularities [Kaw84]. $A$ self-contained and characteristic-free classification in terms of dual graphs of resolutions is given in [Ale92]. terminal singularities are rational and -factorial. We can now formulate the following: $\log$

${\rm Log}$

$\mathbb{Q}$

Definition 1.1. $A$ normal complete surface $Z$ is called a del Pezzo surface if it has only terminal singularities and the anticanonical divisor $-K_{Z}$ is ample. It has index if all of its singularities are of index $\log$

$\log$

$\leq k$

$\leq k.$

We will use the following notation. If $D$ is a -Weil divisor, $D=$ , then will denote the round-up , and $\{D\}=$ $A$ $D$ the fractional part. divisor is nef if for any curve $C$ one has $D\cdot C\geq 0;D$ is big and nef if in addition $D^{2}>0.$ Below we will frequently use the following generalization of Kodaira’s vanishing theorem. The two-dimensional case is due to Miyaoka [Miy80] and does not require the normal-crossing condition. The higher-dimensional case is due to Kawamata [Kaw82] and Viehweg [Vie82]. $\mathbb{Q}$

$\sum c_{i}C_{i},$

$c_{i}\in \mathbb{Q}$

$\ulcorner D^{\urcorner}$

$\sum^{\ulcorner}c_{i^{\urcorner}}D_{i}$

$\sum\{c_{i}\}C_{i}$

Theorem 1.2 (Generalized Kodaira’s Vanishing theorem). Let

$Y$

be a

be a -divisor on such that (1) $supp\{D\}$ is a divisor with normal crossings; (2) $D$ is big and $nef$

smooth surface and let

Then

1.2.

$D$

$H^{i}(K_{Y}+\ulcorner D^{\neg})=0$

${\rm Log}$

$Y$

$\mathbb{Q}$

for $i>0.$

terminal singularities of index 2

Let $(Z,p)$ be a two-dimensional , and terminal singularity of index : be its minimal resolution. We have where-l and $F_{i}^{2}\leq-2$ . Therefore, for each one has $\alpha_{i}=-1/2$ or . One can rewrite the set of equations in a matrix form: $\log$

$\leq 2$

$\tilde{Z}\rightarrow Z$

$K_{\tilde{Z}}=\pi^{*}K_{Z}+\sum\alpha_{i}F_{i},$

$\pi$

$0$ , we may assume that . Now, let us check that, if the positive number is sufficiently small, then $C\cdot T>0$ for any irreducible $(F_{i}\cdot F_{j})$

$\beta_{i}0.$ If $C^{2}0.$

The contradiction thus obtained completes the proof of the theorem.

$\square $

Remark 1.7. In the same way, parts 1 and 3 can be proved for a del $system-\pi^{*}(nK_{Z})$ Pezzo surface of arbitrary index and the linear . Part 2 $\pi(E)$ passes only through (some is easy to prove under the assumption that of) the Du Val singularities. $\log$

$n$

1.5. Reduction to DPN surfaces of elliptic type Let $Z$ be a . Consider the resolution of del Pezzo surface of index $Y\rightarrow Z$ for which every Du Val singularity is resolved singularities : by inserting the usual tree of $(-2)$ -curves, and the singularity by the following chain: $\leq 2$

$\log$ $f$

$K_{n}$

(9)

2n-l vertices

The latter resolution is obtained by blowing up all intersection points of exceptional curves on the minimal resolution of points, see their diagrams in Section 1.2. In contrast to the minimal resolution, we will call this the right resolution of singularities. Consider a smooth element $C_{g}\in|-2K_{Z}|.$ It does not pass through singularities of the surface $Z$ . If we identify the curve with its image under the morphism , then it is easy to see from the formulae of Section 1.2 $that-f^{*}2K_{Z}$ is linearly equivalent to , and $-2K_{Y}$ with the disjoint union of and curves in the above diagrams which $K_{n}$

$f$

$C_{g}$

$C_{g}$

$C_{g}$

19

1.5. REDUCTION TO DPN SURFACES OF ELLIPTIC TYPE

have self-intersection-4. Moreover, it is easy to compute the genus of the curve , and it equals $g=K_{Z}^{2}+1\geq 2$ . This shows that the surface $Y$ is a right DPN surface of elliptic type in the sense of the next Chapter (see Sections 2.1 and 2.8). Vice versa, the results of Chapters 2 and 3 will imply (see Chapter 4) that a right DPN surface $Y$ of elliptic type admits a unique contraction of exceptional curves : to a del Pezzo surface of index In this way, the classification of log del Pezzo surfaces of index is reduced to classification of right DPN surfaces of elliptic type. $C_{g}$

$f$

$Y\rightarrow Z$

$\log$

$\leq 2.$

$\leq 2$

2 General Theory of DPN surfaces and K3 surfaces with non-symplectic involution

2.1. General remarks del Pezzo surfaces of As it was shown in Chapter 1, a description of is reduced to a description of rational surfaces $Y$ containing a index nonsingular curve $C\in|-2K_{Y}|$ and a certain configuration of exceptional curves. Such surfaces $Y$ and exceptional curves on them were studied in the papers [Nik79, Nik83, Nik84a, Nik87] of the second author. They are one of possible generalizations of del Pezzo surfaces. Many other generalizations of del Pezzo surfaces were proposed, see e.g. Dem80, Har85a, Har85b, Loo81], and most authors call their surfaces ”generalized del Pezzo surfaces”. Therefore, we decided following [Nik87] to call our generalization DPN surfaces. One can consider DPN surfaces to be some appropriate non-singular models of log del Pezzo surfaces of index and some their natural generalizations. Definition 2.1. $A$ nonsingular projective algebraic surface $Y$ is called a DPN surface if its irregularity $q(Y)=0,$ $K_{Y}\neq 0$ and there exists an effective divisor $C\in|-2K_{Y}|$ with only simple rational, i.e. $A,$ $D,E$ singulanities. Such a pair $(Y, C)$ is called a DPN pair. A DPN surface $Y$ is called right if there exists a nonsingular divisor $C\in|-2K_{Y}|$ ; in this case the pair $(Y, C)$ is called right DPN pair or nonsingular DPN pair. $\log$

$\leq 2$

$[$

$\leq 2$

-

The classification of algebraic surfaces implies that if $ C=\emptyset$ then a then $Y$ DPN surface $Y$ is an Enriques surface $(x=p=q=0)$ . If is a rational surface $(x=-1, p=q=0)$ , e.g. see [Shaf65]. Using the well-known properties of blowups, the following results are easy to prove. Let $(Y, C)$ be a DPN pair, $E\subset Y$ be an exceptional curve of be the contraction of $E$ . Then $(Y’, \sigma(C))$ the lst kind on $Y$ and is also a DPN pair. In this way, by contracting exceptional curves of the lst kind, one can always arrive at a DPN pair $(Y’, C’)$ where $Y’$ is a relatively minimal (i.e. without exceptional curves of the lst kind) rational surface. In $ C\neq\emptyset$

$\sigma:Y\rightarrow Y’$

20

2. 1. GENERAL REMARKS

21

this case, the only possibilities for $Y’$ are and , since only $|-2K_{Y’}|$ for them contains a reduced divisor. If then $C’$ is a curve of degree 6; if then $C’$ is a curve of bidegree (4, 4); if then $C’\in|8f+4s_{2}|$ ; if then $D’=C_{1}+s_{3}$ , where $C_{1}\in|10f+3s_{3}|$ ; if then $C’=D_{1}+s_{4}$ , where $D_{1}\in|12f+3s_{4}|.$ Here, the linear system is a pencil of rational curves on surface with $P$ $s_{n}^{2}=-n$ C’)$ $(Y’, a section . Vice versa, if is a DPN pair, is a singular $C’$ $P$ point of and : is a blowup of with an exceptional $(-1)-$ curve $E$ then $(Y, C)$ is a DPN pair, where $\mathbb{P}^{2},$

$\mathbb{F}_{0},$

$\mathbb{F}_{2},$

$\mathbb{F}_{4}$

$\mathbb{F}_{3}$

$Y’=\mathbb{P}^{2}$

$Y’=\mathbb{F}_{0}=\mathbb{P}^{1}\times \mathbb{P}^{1}$

$Y’=\mathbb{F}_{2}$

$Y’=\mathbb{F}_{3}$

$Y’=\mathbb{F}_{4}$

$|f|$

$\mathbb{F}_{n}$

$s_{n},$

$Y\rightarrow Y’$

$\sigma$

$C=\left\{\begin{array}{ll}\sigma_{*}^{-1}(C’) & if P has multiplicity 2 on C’\\\sigma_{*}^{-1}(C’)+E & if P has multiplicity 3 on C’\end{array}\right.$

denotes the proper preimage (or the strict transform) of $C’,$ is the closure of the set-theoretic preimage $\sigma^{-1}(C’-\{P\})$ in $Y.$ In this way, by blowups, from an arbitrary DPN pair $(Y’, C’)$ one can always pass to a right DPN pair $(Y, C)$ , i.e. with a nonsingular $C.$ $A$ description of arbitrary DPN pairs and surfaces is thus reduced to a description of right (or nonsingular) DPN pairs $(Y, C)$ , and to right DPN surfaces $Y$ and exceptional curves on them. Here, a curve $E\subset Y$ is called exceptional if $E$ is irreducible and $E^{2}0$ . It follows that

$S_{X}$

NEF( $X$ )

(14)

$=\{x\in\overline{V^{+}(X)}|x\cdot P(X)\geq 0\}$

where $P(X)\subset S_{X}$ denotes the set of all divisor classes of irreducible nonsingular rational ( . all exceptional) curves on $X.$ Let NEF (X) be a hyperplane section. By Riemann-Roch Theorem above, $f\in S_{X}$ with $f^{2}=-2$ is effective if and only if $f>0$ . It $X$ $V^{+}(X))$ for the group follows that NEF( ) is a fundamental chamber in $W^{(2)}(S_{X})$ generated by reflections in all elements $f\in S_{X}$ with $f^{2}=-2.$ Each such gives a reflection $s_{f}\in O(S_{X})$ where $i.$

$e$

$ h\in$

$h$

$($

$f$

(15)

$s_{f}(x)=x+(x\cdot f)f, x\in S_{X},$

in particular, $s_{f}(f)=-f$ and is identical on Since all $F\in P(X)$ have $F^{2}=-2$ , the nef cone NEF( $X$ ) is locally finite in $V^{+}(X)$ , all its faces of codimension one are orthogonal to elements of $P(X)$ . This gives one-to-one correspondence between the faces of codimension one of NEF( $X$ ) and the elements of $P(X)$ . Indeed, let be a codimension one face of NEF (X). Assume $F\in P(X)$ is orthogonal to , containing belongs to the edge of the half-space $F\cdot x\geq 0,$ $f^{\perp}.$

$s_{f}$

$a$

$\gamma$

$i.$

$\gamma,$

$e.$

$x\in S_{X}\otimes \mathbb{R}$

$\gamma$

24

2. DPN SURFACES AND K3 SURFACES WITH INVOLUTION

NEF( $X$ ). Such $F\in S_{X}$ with $F^{2}=-2$ is obviously unique because any . We have element $f\in S_{X}$ which is orthogonal to is evidently , and $F$ is distinguished by the condition $F^{2}=$ $-2$ . In such a way, all faces of codimension one of NEF( $X$ ) give a subset $P(X)’\subset P(X)$ of elements of $P(X)$ which are orthogonal to codimension one faces of NEF( $X$ ). Let us show that $P(X)’=P(X)$ . Obviously it will be enough to show that for any $E\in P(X)$ , the orthogonal projection of NEF( $X$ ) into hyperplane belongs to NEF( $X$ ). The projection is given by the formula $H\mapsto\tilde{H}=H+(H\cdot E)E/2$ for $ H\in$ NEF( $X$ ). Let us show NEF( $X$ ). Let $C$ be an irreducible curve on $X$ . If $C\neq E$ , then that $C\cdot\tilde{H}=C\cdot H+(H\cdot E)(C\cdot E)/2\geq 0$ because $H$ is nef and $C$ is different from $E$ . If $C=E$ , then $C\cdot\tilde{H}=E\cdot\tilde{H}=E\cdot H+(H\cdot E)(E^{2})/2=0\geq 0.$ Thus, NEF( $X$ ). Therefore, we obtain a group-theoretic description of the nef cone of $X$ and all exceptional curves of $X$ : NEF( $X$ ) is the fundamental chamber for the reflection group $W^{(2)}(S_{X})$ acting on $V^{+}(X)$ , this chamber is distinguished by the condition that it contains a hypelplane section of $X$ . The set $P(X)$ of all exceptional curves on $X$ consists of all elements $f\in S_{X}$ which have $f^{2}=-2$ and which are orthogonal to codimension one faces NEF $(X)\geq 0)$ . of NEF( $X$ ) and directed outwards It is more convenient to work with the corresponding hyperbolic space $\lambda F,$

$\gamma$

$\lambda\in \mathbb{R}^{+}$

$(\lambda F)^{2}=\lambda^{2}F^{2}=\lambda^{2}(-2)$

$E^{\perp}$

$\tilde{H}\in$

$\tilde{H}\in$

$(i.$

(16)

$e.$

$ f\cdot$

$\mathcal{L}(X)=V^{+}(X)/\mathbb{R}^{+}$

Elements of this space are rays $x\cdot h>0$ . Each element

$\mathbb{R}^{+}x$

, where

$\beta\in S_{X}\otimes \mathbb{R}$

and defines a half-

$x\in S_{X}\otimes \mathbb{R},$

with square

$\beta^{2}0$

space (17)

$\mathcal{H}_{\beta}^{+}=\{\mathbb{R}^{+}x\in \mathcal{L}(X)|\beta\cdot x\geq 0\},$

so that is perpendicular to the bounding hyperplane $\beta$

(18)

$\mathcal{H}_{\beta}=\{\mathbb{R}^{+}x\in \mathcal{L}(X)|\beta\cdot x=0\},$

and faces outward. The set

$\iota$

(19) $\mathcal{M}(X)=NEF(X)/\mathbb{R}^{+}=\bigcap_{f\in s_{X},J^{2}=-2 fiseffective}\mathcal{H}_{f}^{+}=\bigcap_{f\in P(X)}\mathcal{H}_{f}^{+}$

. The set $P(\mathcal{M}(X))$ of vectors is a locally finite convex polytope in and directed outwith square $-2$ , perpendicular to the facets of ward, is exactly the set $P(X)$ of divisor classes of exceptional curves on $X.$ Moreover, admits a description in terms of groups. Let $O’(S_{X})$ be the subgroup of index two of the full automorphism group $O(S_{X})$ of the latwhich consists of automorphisms preserving the half-cone $V^{+}(X)$ . tice $\mathcal{L}(X)$

$\mathcal{M}(X)$

$\mathcal{M}(X)$

$S_{X}$

25

2.2. REMINDER OF BASIC FACTS ABOUT K3 SURFACES

Let

be the subgroup of $O’(S_{X})$ generated by reflections with respect to all elements $f\in S_{X}$ with square $(-2)$ . The action of the group $W^{(2)}(S_{X})$ , as well as $O’(S_{X})$ , on is discrete. $W^{(2)}(S_{X})$ $f\in S_{X}$ and is the group generated by reflections in all hyperplanes $f^{2}=-2$ . The set is a fundamental chamber for this group, i.e. $W^{(2)}(S_{X})(\mathcal{M}(X))$ defines a decomposition of into polytopes which $W^{(2)}(S_{X})$ , and are congment to acts transitively and without fixed [PS-Sh71, Vin85]). The fundamental elements on this decomposition (cf. chamber is distinguished from other fundamental chambers by the fact that it contains the ray of polarization. By Hodge decomposition, we have the direct sum $W^{(2)}(S_{X})\subset O’(S_{X})$ $s_{f}$

$\mathcal{L}(X)$

$\mathcal{H}_{f},$

$\mathcal{M}(X)$

$\mathcal{L}(X)$

$\mathcal{M}(X)$

$\mathcal{M}(X)$

$\mathbb{R}^{+}h$

(20)

where

$H^{2}(X, \mathbb{Z})\otimes \mathbb{C}=H^{2}(X, \mathbb{C})=H^{2,0}(X)+H^{1,1}(X)+H^{0,2}(X)$

$H^{2,0}(X)=\mathbb{C}\omega_{X},$ $H^{0,2}(X)=\overline{H^{2,0}(X)}$

$H^{0,2}(X))^{\perp}$

(21)

. Then the Picard lattice of

$X$

and

$H^{1,1}(X)=(H^{2,0}(X)+$

is

$S_{X}=H^{2}(X, \mathbb{Z})\cap H^{1,1}(X)=\{x\in H^{2}(X, \mathbb{Z})|x\cdot H^{2,0}(X)=0\}.$

The triplet (22)

$(H^{2}(X, \mathbb{Z}), H^{2,0}(X), \mathcal{M}(X))$

is called the periods of $X.$ An isomorphism (23) $\phi:(H^{2}(X, \mathbb{Z}), H^{2,0}(X), \mathcal{M}(X))\rightarrow(H^{2}(X’, \mathbb{Z}), H^{2,0}(X’), \mathcal{M}(X’))$

of periods of two K3 surfaces means an isomorphism : of cohomology lattices ( . modules with pairing) such that $\phi(H^{2,0}(X))=H^{2,0}(X’),$ $\phi(\mathcal{M}(X))=\mathcal{M}(X’)$ for the corresponding induced maps which we denote by the same letter . By Global Torelli The$\phi$

$H^{2}(X’, \mathbb{Z})$

$i.$

$ H^{2}(X, \mathbb{Z})\rightarrow$

$e$

$\phi$

orem for K3 $f$

:

$X’\rightarrow X$

[PS-Sh71], surraces of the K3 surfaces:

$\phi$

is defined by a unique isomorphism

$\phi=f^{*}$

As an application ofthe Global Torelli Theorem, let us consider the description of Aut(X) from [PS-Sh71]. By Serre duality, $h^{1,2}=0$ . Thus, $X$ has no regular vector-fields. It follows, that Aut(X) acts on with only a finite kemel. Let

$h^{0}(\mathcal{T}_{X})=h^{2}(\Omega_{X}^{1})=$

$S_{X}$

Sym $(\mathcal{M}(X))=\{\phi\in O’(S_{X})|\phi(\mathcal{M}(X))=\mathcal{M}(X)\}$

(24)

be the symmetry group of the fundamental chamber . Let us denote by Sym its subgroup of finite index which consists of all symmetries which are identical on the discriminant group $(S_{X})^{*}/S_{X}$ . EleSym can be extended to automolphisms of ments which are identical on the transcendental lattice (see Propositions A.3, A.4 in Appendix). We denote this extension by $\mathcal{M}(X)$

$(\mathcal{M}(X))^{0}$

$\phi\in$

$(\mathcal{M}(X))^{0}$

$H^{2}(X, \mathbb{Z})$

$T_{X}=(S_{X})^{\perp}\subset H^{2}(X, \mathbb{Z})$

26

2. DPN SURFACES AND K3 SURFACES WITH INVOLUTION

the same letter

$\phi$

since it is unique. We have

.

Thus, phism of periods of . Thus, $H^{2,0}(X)\cdot S_{X}=0$

$H^{2,0}(X)\subset T_{X}\otimes \mathbb{C}$

$X$

$\phi=f^{*}$

where

$ f\in$

since

and is an automorAut(X). Thus, the natural

$\phi(H^{2,0}(X))=H^{2,0}(X)$ ,

$\phi$

contragradient representation

Aut

(25)

$(X)\rightarrow$

Sym

$(\mathcal{M}(X))$

has a finite kemel and a finite cokemel. It follows that the groups Aut $(X)\approx$ Sym are naturally isomorphic up to finite groups. Since we have a natural isomorphism Sym $(\mathcal{M}(X))\cong O’(S_{X})/W^{(2)}(S_{X})$ , we also obtain that $(\mathcal{M}(X))$

Aut $(X)\approx O’(S_{X})/W^{(2)}(S_{X})$ .

(26)

In particular, Aut(X) is finite if and only if $[O(S_{X}) : W^{(2)}(S_{X})]0$ ). Abstract K3 periods is a triplet $\omega_{X}\cdot\overline{\omega_{X}}>0$

(27)

$ 0\neq\omega_{X}\in$

$(L_{K3}, H^{2,0}, \mathcal{M})$

is an even unimodular lattice of signature (3, 19), is a one dimensional complex linear subspace satisfying $H^{2,0}\cdot H^{2,0}=0,$ , and is a fundamental chamber of $W^{(2)}(M)\subset \mathcal{L}(M)$ where $M=\{x\in L_{K3}|x\cdot H^{2,0}=0\}$ is an abstract Picard lattice. By the surjectivity of Torelli map for $K3$ surfaces [Ku177], any abstract K3 periods are isomorphic to periods of an algebraic K3 surface. As an application of Global Torelli Theorem and Surjectivity of Torelli map for K3 surfaces, let us describe moduli spaces ofK3 surfaces with condition on Picard lattice. For details see [Nik80a] and for real case [Nik84b]. is even for any $x\in M$ ) hyperbolic (i. e. of Let $M$ be an even ( i. e. above, we consider the light signature $(1, rk M-1)$ ) lattice. Like for where

$ H^{2,0}\subset L_{K3}\otimes$

$L_{K3}$

$\mathbb{C}$

$H^{2,0}\cdot\overline{H^{2,0}}>0$

$\mathcal{M}$

$x^{2}$

$S_{X}$

cone (28)

$V(M)=\{x\in M\otimes \mathbb{R}|x^{2}>0\}$

of $M$ , and we choose one of its half $V^{+}(M)$ defining the corresponding hyperbolic space $\mathcal{L}(M)=V^{+}(M)/\mathbb{R}^{+}$ . We choose a fundamental chamber for the reflection group $W^{(2)}(M)$ generated by reflections in all elements $f\in M$ with $f^{2}=-2$ . Note that the group $\pm W^{(2)}(M)$ acts transitively on all these additional data $(V^{+}(M), \mathcal{M}(M))$ which shows that they are defined by the lattice $M$ itself (i. e. by its isomorphism class), and we can fix these additional data $(V^{+}(M), \mathcal{M}(M))$ without loss of generality. $\mathcal{M}(M)\subset \mathcal{L}(M)$

27

2.2. REMINDER OF BASIC FACTS ABOUT K3 SURFACES

such that a primitive sublattice $M\subset S_{X}$ is fixed, $V^{+}(X)\cap(M\otimes \mathbb{R})=V^{+}(M),$ , and . (This is one of the weakest possible conditions of degeneration.) Such a K3 surface $X$ is called a K3 surface with the condition $M$ on Picard lattice. $A$ general K3 surface $X$ with the condition $M$ on Picard lattice ( . with moduli or periods general enough) has $S_{X}=M$ , as we will show later. Then $S_{X}=M,$ $V^{+}(X)=V^{+}(M)$ , and $\mathcal{M}(X)=\mathcal{M}(M)$ . One can consider this condition as a marking of elements of the Picard lattice by elements of the standard lattice $M.$ Let $(X, M\subset S_{X})$ be a K3 surface with the condition $M$ on the Picard lattice. Then $M\subset S_{X}\subset H^{2}(X, \mathbb{Z})$ defines a primitive sublattice . Depending on the isomorphism class of this primitive sublattice, we obtain different irreducible components of moduli of K3 surfaces with the condition $M$ on Picard lattice. We fix a primitive embedding $M\subset L_{K3}$ . We consider marked K3 surfaces $(X, M\subset S_{X})$ with the condition $M$ on Picard lattice and the class $M\subset L_{K3}$ of the condition $M$ on cohomology. Here marking means an isomorphism : $H^{2}(X, \mathbb{Z})\cong L_{K3}$ of lattices such that $\xi|M$ is identity. Taking

We consider K3 surfaces

$X$

$\mathcal{M}(X)\cap \mathcal{L}(M)\neq\emptyset$

$\mathcal{M}(X)\cap$

$\mathcal{L}(M)\subset \mathcal{M}(M)$

$i.$

$e$

$S_{X}$

$ M\subset$

$H^{2}(X, \mathbb{Z})$

$\xi$

(29)

$(L_{K3}, H^{2,0}=\xi(H^{2,0}(X)), \mathcal{M}=\xi(\mathcal{M}(X)))$

we obtain periods of a marked K3 surface $(X, M\subset S_{X}, \xi)$ with condition $M$ on Picard lattice. By the surjectivity of Torelli map, any abstract periods $(L_{K3}, H^{2,0}, \mathcal{M})$

, and where $H^{2,0}\cdot M=0,$ correspond to a marked K3 surface with the condition $M$ on Picard lattice. Let us denote by the space of all these abstract periods. It is called the period domain of K3 surfaces $(X, M\subset S_{X})$ with the condition $M$ on Picard lattice and with the type $M\subset L_{K3}$ of this condition on cohomology. Let $\mathcal{M}\cap \mathcal{L}(M)\neq\emptyset$

$\mathcal{M}\cap \mathcal{L}(M)\subset \mathcal{M}(M)$

$\tilde{\Omega}_{M\subset L_{K3}}$

(30) $\Omega_{M\subset L_{K3}}=\{H^{2,0}=\mathbb{C}\omega\subset L_{K3}\otimes \mathbb{C}|\omega\cdot M=0,$

$\omega^{2}=0$

and

$\omega\cdot\overline{\omega}>0\}.$

We have the natural projection : forgetting The space is an open subset of a projective quadric of dimension $L_{K3}-$ rk rk $M-2=20-$ rk $M$ . It follows that for a general K3 surface $X$ with the condition $M$ on Picard lattice we have $S_{X}=M$ . Indeed, if rk $S_{X}>$ rk $M$ for all K3 surfaces $X$ with the condition $M\subset L_{K3}$ , then, since $H^{2,0}\cdot\xi(S_{X})=0$ , periods define a quadric of smaller dimension 20-rk $S_{X}0$

$S$

$S_{X}$

$S.$

$S\subset S_{X}$ $S$

$(S_{X})_{-}$

$S_{X}$

$f$

$\pm f$

$S$

$H^{2}(X, \mathbb{Z})\rightarrow S^{*}$

$H^{2}(X, \mathbb{Z})$

$H^{2}(X, \mathbb{Z})\rightarrow\tau*$

$\theta$

$S^{*}/S\cong H^{2}(X, \mathbb{Z})/(S\oplus T)\cong T^{*}/T$

30

2. DPN SURFACES AND K3 SURFACES WITH INVOLUTION

is an unimodular lattice. The involution is $+1$ on $S^{*}/S,$ because and it is-l on $\tau*/T$ . It follows that the groups coincide. Thus, are 2-elementary. Only in this case multiplications by the lattice is 2-elementary, which means that its discriminant group is 2-elementary where gives another important invariant of There is one more invariant of which takes values in $\{0,1\}$ . One the discriminant for every has quadratic form $\theta$

$H^{2}(X, \mathbb{Z})$

$S^{*}/S\cong T^{*}/T\cong(\mathbb{Z}/2\mathbb{Z})^{a}$ $\pm 1$

$S$

$\mathfrak{U}s=$

$S^{*}/S\cong(\mathbb{Z}/2\mathbb{Z})^{a}$

$a$

$S.$

$\delta$

$\delta=0\Leftrightarrow$

$S$

$ x^{*}\in S^{*}\Leftrightarrow$

$(x^{*})^{2}\in \mathbb{Z}$

$q_{S}:\mathfrak{U}_{S}=S^{*}/S\rightarrow \mathbb{Q}/2\mathbb{Z}, q_{S}(x^{*}+S)=(x^{*})^{2}+2\mathbb{Z}$

. See Appendix, Section A. 1 about of is even: it takes values in discriminant forms of lattices. The invariants of define the isomorphism class ofa 2-elementary even hyperbolic lattice . See more general statement and the proof in Appendix, Section A.2 and Theorem A.9. Thus, any two even hyperbolic 2-elementary lattices with the same inof are equivalent variants are isomorphic. The invaniants to the main invariant , and we later call them the main invariants of a K3 surface $X$ with non-symplectic involution Vice versa, let be a hyperbolic even 2-elementary lattice having a . Let be one of primitive embeddings. primitive embedding to $T=S^{\perp}$ in and the diagram similar to above, Considering $S$

$(\mathbb{Z}/2\mathbb{Z})$

$S$

$(r, a, \delta)$

$S$

$(r, a, \delta)$

$(r, a, \delta)$

$S$

$S$

$\theta.$

$S$

$S\subset L_{K3}$

$L_{K3}$

$L_{K3}$

$S^{*}/S\cong L_{K3}/(S\oplus T)\cong T^{*}/T,$

we obtain that there exists an involution of $-1$ on . Let us consider the moduli space $\theta^{*}$

$L_{K3}$

which is

$+1$

on

$S$

, and

$T$

(35)

$Mod_{s}’\subset Mod_{S}$

of K3 surfaces $(X, S\subset S_{X})$ with condition on Picard lattice (see (34)) where for $(X, S\subset S_{X})$ from $Mod_{s}’$ we additionally assume that the orhas no elements with square $(-2)$ . to in thogonal complement One can easily see that $Mod_{S}’$ is Zariski open subset in $Mod_{S}$ . Any general $(X, S\subset S_{X})(i.$ . when $S=S_{X})$ belongs to $Mod_{s}’$ . Thus, the difference between $Mod_{s}’$ and $Mod_{S}$ is in complex codimension one, and they have the same irreducible components. By Global Torelli Theorem, the accan be lifted to a non-symplectic involution on $X$ with tion of on $H^{2}(X, \mathbb{Z})_{+}=S$ . Thus, the moduli space $Mod_{s}’$ in (35) can be considered as moduli space of K3 surfaces with non-symplectic involution and the , it can main invariant . Since is defined by the main invariants also be denoted as $S$

$S^{\perp}$

$S$

$S_{X}$

$e$

$\theta^{*}$

$S$

(36)

$\theta$

$L_{K3}$

$S$

$(r, a, \delta)$

$Mod_{(r,a,\delta)}=Mod_{S}’$

2.3. THE LATTICE

$S$

,

AND THE MAIN INVARIANTS

$(r, a, \delta),$

FIGURE 1. All possible main invariants

31

$(k, g, \delta)$

$(r, a, \delta)$

and can be considered as moduli space of K3 surfaces with non-symplec. Any even hyperbolic 2tic involution and the main invariants (up to isoelementary lattice has a unique primitive embedding to morphisms) if it exists. Moreover, the group $O(S\subset L_{K3})$ always has an is automorphism of spinor norm $-1$ . Thus, the moduli space Mod irreducible. Evidently, to classify all possible main invariants (equivalently one just needs to classify all even hyperbolic 2-elementary lattices having a primitive embedding $S\subset L_{K3}$ . All such possibilities for (equivalently,$ (k=(r-a)/2,$ $g=(22-r-a)/2,$ , see below) are known and are shown on Figure 1. The triple admits an interpretation in terms of $X^{\theta}=C$ . If $(r, a, \delta)\neq(10,8,0)$ or $(10, 10, 0)$ then $(r, a, \delta)$

$S$

$L_{K3}$

$(r,a,\delta)$

$S$

$(r,$ $a,$

$S$

$\delta))$

$(r, a, \delta)$

$\delta)$

$(r, a, \delta)$

$X^{\theta}=C=C_{g}+E_{1}+\cdots+E_{k},$

are is a nonsingular irreducible curve of genus , and where nonsingular irreducible rational curves, the curves are disjoint to each other, $C_{g}$

$g$

$E_{1},$

$\ldots E_{k}$

$g=(22-r-a)/2, k=(r-a)/2$ (we shall formally use the same formulae for and even in cases$ (r, a, \delta)=$ $(10,8,0)$ or $(10, 10, 0))$ . If $(r, a, \delta)=(10,8,0)$ then $X^{\theta}=C=C_{1}^{(1)}+C_{1}^{(2)},$ are elliptic (genus 1) curves. If $(r, a, \delta)=(10,10,0)$ then where , i.e. in this case $Y$ is an Enriques surface. One has $ g $

$k $

$C_{1}^{(i)}$

$ X^{\theta}=C=\emptyset$

(37) $\delta=0\Leftrightarrow X^{\theta}\sim 0$

$mod 2$

in

$S_{X}$

(equivalently in

$ H_{2}(X, \mathbb{Z})$

)

$\Leftrightarrow$

32

2. DPN SURFACES AND K3 SURFACES WITH INVOLUTION

there exist signs (38)

for which

$(\pm)_{i}$

$\frac{1}{4}\sum_{i}(\pm)_{i}d(C^{(i)})\in S_{Y}=H^{2}(Y, \mathbb{Z})$

,

go over all irreducible components of $C$ . Signs for $\delta=0$ are defined uniquely up to a simultaneous change. They define a new natural orientation (different from the complex one) of the components of $C$ ; a positive $sign$ gives the complex orientation and a negative $sign$ the opposite orientation. The main invariants , equivalently of K3 surfaces with non-symplectic involution, and the corresponding DPN pairs and DPN surfaces play a cmcial role in our classification. where

$C^{(i)}$

$(\pm)_{i}$

$S$

2.4. Exceptional curves on

$(r, a, \delta)$

$(X, \theta)$

$(or (k, g, \delta))$

and

$Y$

A description of exceptional curves on a DPN surface $Y$ can also be reduced to the K3 surface $X$ with a non-symplectic involution considered above. Let be a K3 surface with a non-symplectic involution and $(Y=$ $C=\pi(X^{\theta}))$ the corresponding DPN pair. If $E\subset Y$ is an exceptional curve, then the curve $F=\pi^{*}(E)_{red}$ is either an irreducible curve with negative square on the K3 surface $X$ , or $F=F_{1}+\theta(F_{1})$ , where $\theta$

$(X, \theta)$

$X/\{1, \theta\},$

$F^{2}=F_{1}^{2}+\theta(F_{1})^{2}+2F_{1}\cdot\theta(F_{1})=2E^{2}2 $is small), if

$\frac{2f_{1}\cdot f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}}=2\cos\frac{\pi}{m}, m\in N.$

They are connected by a thick edge, if

$\frac{2f_{1}\cdot f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}}=2.$

They are connected by a broken edge of the weight , if $t$

$\frac{2f_{1}\cdot f_{2}}{\sqrt{f_{1}^{2}f_{2}^{2}}}=t>2.$

Moreover,

a vertex corresponding to

is black. It is transparent, if . It is double transparent, if $f\in P(X)_{+I}(i.$ . it corresponds to the class of a rational component of ), otherwise, it is simple transparent. Of course, here we assume that for a K3 surface with involution and $(S_{X})_{+}=S.$ Classification of DPN surfaces of elliptic type is based on the purely arithmetic calculations of the fundamental chambers (equivalently, $W^{(2,4)}(S)$ of the graphs of the reflection groups of the lattices $W^{(2,4)}(S)=W(S)$ even, of elliptic type. Since is -elementaly and is the full reflection group of the lattice , and any root $f\in S$ has $f^{2}=-2$ $or-4$ . We have $f\in P^{(4)}(\mathcal{M}^{(2,4)})$

$f\in P^{(2)}(\mathcal{M}^{(2,4)})$

$X^{\theta}$

$e$

$\mathcal{M}^{(2,4)}\subset \mathcal{M}(X)_{+}$

$(X, \theta)$

$\mathcal{M}^{(2,4)}$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$S$

$S$

$2$

$S$

Theorem 3.1. 2-elementary even hyperbolic lattices of elliptic type have fundamental chambers for their reflection gmups $W^{(2,4)}(S)$ (it is the full reflection gmup of ), equivalently the corresponding Dynkin diagrams , which are given in Table 1 below, where the lattice is ), see Section 2.3. (equivalently, defined by its invariants $S$

$\mathcal{M}^{(2,4)}$

$S$

$S$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$(r, a, \delta)$

$(k, g, \delta)$

52

3. DPN SURFACES OF ELLIPTIC TYPE

of reflection TABLE 1. Fundamental chambers groups $W^{(2,4)}(S)$ for 2-elementaly even hyperbolic lattices of elliptic type. $\mathcal{M}^{(2,4)}$

$S$

3.1. FUNDAMENTAL CHAMBERS OF

$W^{(2,4)}(S)$

FOR ELLIPTIC TYPE

53

54

3. DPN SURFACES OF ELLIPTIC TYPE

3.1. FUNDAMENTAL CHAMBERS OF

$W^{(2,4)}(S)$

FOR ELLIPTIC TYPE

55

Proof. When is unimodular $(i.e. a=0)$ or $r=a$ (then $S(1/2)$ is unimodular), i. e. for cases 1–11, 40, 50, these calculations were done by Vnberg [Vin72]. In all other cases they can be done using Vinberg’s algorithm for calculation of the fundamental chamber of a hyperbolic reflection group. See [Vin72] and also [Vin85]. These technical calculations take too much space and will be presented in Appendix, Section A.4.1. To describe elements of $P(X)_{+I}$ ( . double transparem vertices), we use the results of Section 2.6 and the fact that their number is known by Section 2.3. $S$

$i.$

$e$

$k$

$\square $

Remark 3.2. Using diagrams of Theorem 3.1, one can easily find the class as an element $C_{g}\in S$ such that $C_{g}\cdot x=0,$ of in of the component if corresponds to a black or a double transparent vertex, and $C_{g}\cdot x=2-s$ if corresponds to a simple transparent vertex which has edges to double transparent vertices. $S$

$X^{\theta}$

$C_{g}$

$x$

$x$

$s$

56

3. DPN SURFACES OF ELLIPTIC TYPE

3.2. Root invariants, and subsystems of roots in for elliptic case $\Delta^{(4)}(M^{(2)})$

We use the notation and results of Section 2.4.1. Let be $W^{(2)}(S)$ the fundamental chamber of . Dynkin diagram containing ( . black vertices) consists of components of types $A,$ $D$ of or $E$ (see Table 1). Thus, the group generated by reflections in all elements of is a finite Weyl group. It has to be finite has finite volume, and because is the in . Thus, fundamental chamber for the action of $\mathcal{M}^{(2)}\supset \mathcal{M}^{(2,4)}$

$\mathcal{M}^{(2,4)}$

$P^{(4)}(\mathcal{M}^{(2,4)})$

$i.$

$e$

$W^{(4)}(\mathcal{M}^{(2)})$

$P^{(4)}(\mathcal{M}^{(2,4)})$

$\mathcal{M}^{(2,4)}$

$W^{(4)}(\mathcal{M}^{(2)})(\mathcal{M}^{(2,4)})=\mathcal{M}^{(2)}$

$W^{(4)}(\mathcal{M}^{(2)})$

$\mathcal{M}^{(2)}$

$\Delta^{(4)}(\mathcal{M}^{(2)})=W^{(4)}(\mathcal{M}^{(2)})P^{(4)}(\mathcal{M}^{(2,4)})$

is a finite root system of the corresponding type with the negative definite root sublattice $R(2)=[P^{(4)}(\mathcal{M}^{(2,4)})]\subset S.$

be a K3 surface with a non-symplectic involution, and $(S_{X})_{+}=S$ . Let be the subset defined by which (we remind that it is generated by reflecis invaniant with respect to ). By Theorem 2.4, tions in and where is a root subsystem in . Let

Let

$(X, \theta)$

$\Delta_{+}^{(4)}\subset\triangle(4)(S)$

$(X, \theta)$

$W_{+}^{(2,4)}$

$\triangle^{(2)}(S)$

$\Delta_{+}^{(4)}=W^{(2)}(S)\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

$\triangle_{+}^{(4)}$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})=\triangle_{+}^{(4)}\cap\Delta^{(4)}(\mathcal{M}^{(2)})$

(64)

$\triangle(4)(\mathcal{M}^{(2)})$

$K^{+}(2)=[\Delta_{+}^{(4)}(\mathcal{M}^{(2)})]\subset R(2)\subset S$

be its negative definite root sublattice in , and $S$

(65)

$Q=\frac{1}{2}K^{+}(2)/K^{+}(2) , \xi^{+}:q_{K(2)}+|Q\rightarrow q_{S}$

a homomorphism such that

$\xi^{+}(x/2+K^{+}(2))=x/2+S,$

$x\in K^{+}(2)$

.

We obtain a pair $(K^{+}(2), \xi^{+})$ which is similar to a root invariant, and it is equivalent to the root invariant for elliptic type.

be a $K3$ surface with a non-symplectic involuProposition 3.3. Let tion of elliptic type, and $S=(S_{X})_{+}.$ In this case, the root invariant $R(X, \theta)$ is equivalent to the root sub, considered up to the action of $O(S)$ system . two mot subsystems and are equivalent, for some $O(S))$ : The mot invariant $R(X, \theta)\cong(K^{+}(2), \xi^{+})$ is defined by (64) and (65). The fundamental chamber is defined by the mot subsystem (up to above equivalence), by Theorem 2.4. $(X, \theta)$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})\subset\triangle^{(4)}(\mathcal{M}^{(2)})$

$(i.$

$e$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset\triangle^{(4)}(\mathcal{M}^{(2)})$

$\Delta^{(4)}(\mathcal{M}^{(2)})$

$lf\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’=\phi(\triangle_{+}^{(4)}(\mathcal{M}^{(2)}))$

$\mathcal{M}(X)_{+}$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset\Delta^{(4)}(\mathcal{M}^{(2)})$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’\subset$

$\phi\in$

3.2. ROOT INVARIANTS, AND SUBSYSTEMS OF ROOTS IN

Moreover, $\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

Proof. Let

$P^{(4)}(\mathcal{M}(X)_{+})$

coincides with a basis

57

$\Delta^{(4)}(\mathcal{M}^{(2)})$

of the

mot subsystem

. $E_{i},$

$E_{i}\cdot\theta(E_{i})=0,$

be all non-singular rational curves on

$i\in I$ , $i.$

$X$

such that

$e.$

cl $(E)+$ cl $(\theta(E))=\delta\in P^{(4)}(\mathcal{M}(X)_{+})=P^{(4)}(X)_{+}=P(X)_{+III}.$ $i\in I$ , generate in Since $E_{i}\cdot C_{g}=0$ and $C_{g}^{2}=2g-2>0$ , the curves a negative definite sublattice. Thus, their components define a Dynkin diagram which consists of several connected components or The involution acts on these diagrams and corresponding curves without fixed points. Thus it necessarily changes connected components of . Let , and $I=I_{1}uI_{2}$ the corresponding subdiwhere vision of vertices of . Then $E_{i},$

$S_{X}$

$\Gamma$

$A_{n},$

$D_{m}$

$E_{k}.$

$\theta$

$\Gamma$

$\Gamma=r_{1}u\Gamma_{2}$

$\theta(\Gamma_{1})=\Gamma_{2}$

$\Gamma$

$\delta_{i}^{+}=c1(E_{i})+c1(\theta(E_{i})), i\in I_{1},$

and $\delta_{i}^{-}=c1(E_{i})-c1(\theta(E_{i})), i\in I_{1}$

give bases of root systems The map

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

and

$\triangle_{-}^{(4)}=\triangle^{(4)}(K(2))$

respectively.

$\delta_{i}^{-}=c1(E_{i})-c1(\theta(E_{i}))\mapsto\delta_{i}^{+}=c1(E_{i})+c1(\theta(E_{i})), i\in I_{1},$

defines an isomorphism of root systems, since it evidently preserves the intersection pairing. The homomorphism of the root invariant $R(X, \theta)=(K(2), \xi)$ of the pair then goes to $(K^{+}(2), \xi^{+})$ . In the opposite direction, the root invariant $R(X, \theta)$ defines and $\triangle_{-}^{(4)}\cong\Delta_{+}^{(4)}(\mathcal{M}^{(2)})$

$\xi$

$(X, \theta)$

$\triangle_{+}^{(4)}$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})=\triangle^{(4)}(\mathcal{M}^{(2)})\cap\triangle_{+}^{(4)}.$

The last statement follows from Section 2.4.1.

$\square $

By Proposition 3.3, in the elliptic case instead of root invariants one can consider root subsystems . We say that a root in subsystem “is contained” (respectively “is primitively contained”) in a root subsystem , if (respectively is a plimitive embedding of lattices) for some $\phi\in O(S)$ . By Corollary 2.11, we obtain $\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

$($

$\triangle^{(4)}(\mathcal{M}^{(2)}))$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’$

$\phi(\triangle_{+}^{(4)}(\mathcal{M}^{(2)}))\subset\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’$

$[\phi(\triangle_{+}^{(4)}(\mathcal{M}^{(2)}))]\subset[\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’]$

Proposition 3.4. $Ifa$ mot subsystem in corresponds $K3$ , then any primito a surface with non-symplectic involution tive mot subsystem in corresponds to a $K3$ surface with nonsymplectic involution. Thus, it is enough to describe extremal pairs such that their mot subsystems in are not contained as primitive $\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

$\triangle^{(4)}(\mathcal{M}^{(2)})$

$(X, \theta)$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})$

$(X, \theta)$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})$

$\triangle^{(4)}(\mathcal{M}^{(2)})$

3. DPN SURFACES OF ELLIPTIC TYPE

58

mot subsystems

ofstrictly smaller $mnk$ in a mot subsystem

corresponding to another pair

$\Delta^{(4)}(\mathcal{M}^{(2)})$

$(X\prime, \theta^{l})$

in

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’$

.

3.3. Classification of non-symplectic involutions elliptic type of K3 surfaces

$(X, \theta)$

of

We have

and Theorem 3.5. Let $K3$ elliptic type of surfaces. $(X, \theta)$

be two non-symplectic involutions of

$(X\prime, \theta’)$

Then the following three conditions are equivalent: $(equivalently, (k, g, \delta)$) coincide, and (i) Their main invariants their root invariants are isomorphic. coincide, and the mot subsystems (ii) Their main invariants are equivalent. (iii) Dynkin diagrams $\Gamma(P(X)_{+})$ and $\Gamma(P(X’)_{+})$ of their exceptional curves are isomorphic, and additionally the genem are equal, these diagrams are empty. The diagram $\Gamma(P(X)_{+})$ is empty and only ifeither $(r, a, \delta)=(1,1,1)$ then $g=10)$ , or $(r, a, \delta)=(2,2,0)$ (then $g=9$) and or the mot invariant is $zem$ . The corresponding $DPN$ surfaces are $(r, a, \delta)$

$(r, a, \delta)$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})$

$\iota f$

$g$

$\iota f$

$($

$\mathbb{P}^{2}$

$\mathbb{F}_{0}$

respectively. Proof. By Sections 3.2 and 2.5, the conditions (i) and (ii) are equivalent, and they imply (iii). Let us show that (iii) implies (i). Assume that $r=$ rk $S\geq 3.$ , if $r=$ rk $S\geq 3$ . If First, let us show that is generated by $r\geq a+2$ , then it is easy to see that either $S\cong U\oplus T$ or $S\cong U(2)\oplus T$ where $T$ $D_{2m},$ (one can get all possible invariants is orthogonal sum of of taking these orthogonal sums). We have $U=[c_{1}, c_{2}]$ where ). Then is (the same for $U(2)$ , only and generated by elements with square-2 which are $\triangle^{(2)}(S)$

$S$

$A_{1},$

$(r, a, \delta)$

$E_{7},$

$E_{8}$

$S$

$c_{1}^{2}=c_{2}^{2}=0$

$c_{1}\cdot c_{2}=2$

$c_{1}\cdot c_{2}=1$

$\Delta^{(2)}(T)\cup(c_{1}\oplus\Delta^{(2)}(T))\cup(c_{2}\oplus\Delta^{(2)}(T))$

$S$

.

be the corresponding orthogo. Let If $r=a$ then $h^{2}=2$ $e_{i}^{2}=-2,$ $i=1,$ . Then is generated and nal basis of where $h-e_{1}-e_{2}.$ $(-2)$ and which are by elements with square Now, let us show that $P(X)_{+}$ generates . Indeed, every element of can be obtained by composition of reflections in elements of $P(X)_{+}$ from some element of $P(X)_{+}$ . It follows, that it is an integral linear combination of elements of $P(X)_{+}$ . Since we can get in this way all $S\cong\langle 2\rangle\oplus tA_{1}$

$h,$

$e_{1},$

$\ldots,$

$e_{t}$

$S$

$t$

$\ldots,$

$e_{1},$

$\ldots,$

$S$

$\triangle^{(2)}(S)\cup\triangle_{+}^{(4)}$

$e_{t}$

$S$

59

3.3. CLASSIFICATION OF NON-SYMPLECTIC INVOLUTIONS

elements of

$\triangle^{(2)}(S)$

and they generate , it follows that $S$

$P(X)_{+}$

generates

$S.$

It follows that the lattice with its elements $P(X)_{+}$ is defined by the Dynkin diagram $\Gamma(P(X)_{+})$ . From , we can find invariants of and they define invariants . Let $K^{+}(2)\subset S$ be a sublattice generated by $P^{(4)}(X)_{+}(i.$ . by the black vertices), and $\xi^{+}:Q=(1/2)K^{+}(2)/K^{+}(2)\rightarrow q_{S}$ the homomorphism with $\xi^{+}(x/2+K^{+}(2))=x/2+S$ . By Proposition 3.3, the pair $(K^{+}(2), \xi^{+})$ coincides with the root invariant $R(X, \theta)$ . . Then Now assume that $r=$ rk $S=1,2$ for the pair $S$

$S$

$(r, a, \delta)$

$S,$

$(k, g, \delta)$

$e$

$(X, \theta)$

$U(2),$

$U$

or

$S\cong\langle 2\rangle,$

$\langle 2\rangle\oplus\langle-2\rangle.$

In the first two cases and then $ P^{(2)}(X)_{+}=\emptyset$ . In the last two cases and $P^{(2)}(X)_{+}$ are not empty. Thus, only the first two cases give an empty diagram $P^{(2)}(X)_{+}$ . This , the distinguishes these two cases from all others. In the case invariant $g=10$ , and the root invariant is always zero because has no elements with square-4. Thus, in this case, the diagram $P(X)_{+}$ is always empty. This case gives $Y=X/\{1, \theta\}\cong \mathbb{P}^{2}$ . In the case $S=U(2)$ , the diagram $P^{(2)}(X)_{+}$ is empty, but $ P^{(4)}(X)_{+}=\emptyset$ , if the root invariant is zero, and $P^{(4)}(X)_{+}$ consists of one black vertex, if the root invariant is not zero (see Table 1 for this case). First case gives . Second case gives . In both these cases $g=9$ . Thus difference between two cases when the diagram is empty is in genus: $g=10$ for the first case, or and $g=9$ for the second. The difference of $S=U(2)$ with a non-empty diagram $\Gamma(P(X)_{+})$ from all other cases is that this diagram consists of only one black vertex. All cases with rk $S\geq 3$ must have at least 3 different vertices to generate , the diagram $\Gamma(P(X)_{+})$ also consists In cases $S=U$ and of one vertex, but it is respectively double transparent and simple transparent (see Table 1). Moreover, this consideration also shows the difference between cases $S=U$ and and with all other cases. $\triangle^{(2)}(S)=\emptyset$

$\Delta^{(2)}(S)$

$ S=\langle 2\rangle$

$S$

$Y=\mathbb{F}_{0}$

$Y=\mathbb{F}_{2}$

$(\mathbb{P}^{2}$

$\mathbb{F}_{1})$

$S.$

$ S=\langle 2\rangle\oplus\langle-2\rangle$

$\square $

$ S=\langle 2\rangle\oplus\langle-2\rangle$

Theorem 3.5 shows that to classify pairs of elliptic type, we can use any of the following invariants: either the root invariant, or the root ), or the or subsystem (together with the main invariants Dynkin diagram of exceptional curves. It seems that the most natural and geometric is the classification by the Dynkin diagram. Using this diagram, on the one hand, it easy to calculate all other invariants. On the other hand, considering the corresponding DPN surface, we get the Gram diagram of all exceptional curves on it and all possibilities to get the DPN surface by blow-ups from relatively minimal rational surfaces. $(X, \theta)$

$ (k, g, \delta)$

$(r, a, \delta)$

60

3. DPN SURFACES OF ELLIPTIC TYPE

However, the statements (i) and (ii) of Theorem 3.5 are also very imporand tant since they give a simple way to find out if two pairs (equivalently, the corresponding DPN surfaces) have isomorphic Dynkin diagrams of exceptional curves. Moreover, the classification in terms of root invariants and root subsystems is much more compact, since the full Gram diagram of exceptional curves can be very large (e.g. recall the classical non-singular del Pezzo surface corresponding to ). We have the following $(X, \theta)$

$(X\prime, \theta’)$

$E_{8}$

Theorem 3.6 (Classification Theorem in the extremal case of elliptic type). $AK3$ surface with a non-symplectic involution of elliptic type is extremal, and only the number of its exceptional curves with the square $\# P^{(4)}(X)_{+}$ , is equal to $(-4),$ (see Theorem 3.1) where $W^{(2,4)}(S),$ $S=(S_{X})_{+}$ . Equivalently, is a fundamental chamber of $\Gamma(P(X)_{+})$ and numbers ofblack vertices ofDynkin diagrams are equal. with the same invariants $\Gamma(P(X)_{+})$ Moreover, the diagram is isomorphic to ( . coincides with) (see Table 1) in all cases of Theorem 3.1 except cases 7, 8, 9, 10 and20 of Table 1. In the lastfive cases, allpossible diagrams $\Gamma(P(X)_{+})$ are given in Table 2. All diagrams of Tables 1 and 2 correspond to some . extremal standard $K3$ pairs $(X, \theta)$

$\iota f$

$i.$

$\iota f$

$\# P^{(4)}(\mathcal{M}^{(2,4)})$

$e.$

$\mathcal{M}^{(2,4)}$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$(r, a, \delta)$

$i.$

$e$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$(X, \theta)$

Proof. It requires long considerations and calculations and will be given in Section 3.4 below. $\square $

. The Now let us consider a description of non-extremal pairs $\Gamma(P(X)_{+})$ , since the worst way to describe them is using full diagrams $\Gamma(P(X)_{+})$ is very large and diagrams number of non-extremal pairs can be huge. It is better to describe them using Proposition 3.4 and Theoin the root subsystems rem 3.5, by primitive root subsystems . of extremal pairs . By SecLet us choose in such a way that is the subsystem of roots tion 2.4.1, then , i. e. with the basis is the genset of all elements with the square $(-4)$ in the sublattice in $S=(S_{\tilde{X}})_{+}$ . Equivalently, erated by , where is the finite Weyl group generated by reflections in all elements of $P^{(4)}(X)_{+}.$ Replacing a primitive root subsystem by an equivalent root subsystem for a non-extremal pair , we can assume (by primitivity) that a basis $(X, \theta)$

$(X, \theta)$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset\triangle^{(4)}(\mathcal{M}^{(2)})$

$(\tilde{X},\tilde{\theta})$

$\mathcal{M}^{(2)}\supset \mathcal{M}(\tilde{X})_{+}$

$\mathcal{M}^{(2)}$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})=\triangle^{(4)}([P^{(4)}(\tilde{X})_{+}])$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})=\triangle^{(4)}([P^{(4)}(\tilde{X})_{+}])$

$P^{(4)}(\tilde{X})_{+}$

$[P^{(4)}(\tilde{X})_{+}]$

$\triangle^{(4)}([P^{(4)}(\tilde{X})_{+}])=$

$P^{(4)}(\tilde{X})_{+}$

$W_{+}^{(4)}(\tilde{X})(P^{(4)}(\tilde{X})_{+})$

$ W_{+}^{(4)}(\tilde{X})\sim$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’\subset\triangle^{(4)}([P^{(4)}(\tilde{X})_{+}])$

$(X, \theta)$

$\phi(\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’),$

$\phi\in W_{+}^{(4)}(\tilde{X})$

61

3.3. CLASSIFICATION OF NON-SYMPLECTIC INVOLUTIONS

of the root system is a part of the basis . Thus, we can assume that the root subsystem is defined by a subdiagram of

$P^{(4)}(\tilde{X})_{+}$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’$

$\triangle^{(4)}([P^{(4)}(\tilde{X})_{+}])$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’$

$D\subset\Gamma(P^{(4)}(\tilde{X})_{+})$

genwhere is the subdiagram of the full diagram erated by all its black vertices. The $D$ is a basis of By Propositions 2.2, 2.3 and Theorem 2.4, the subdiagram defines the full Dynkin diagram $\Gamma(P(X)_{+})$ of the pair : We have with the root subsystem $\Gamma(P(\tilde{X})_{+})$

$\Gamma(P^{(4)}(\tilde{X})_{+})$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})’.$

$D\subset\Gamma(P^{(4)}(\tilde{X})_{+})$

$(X, \theta)$

(66)

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})’$

$P^{(2)}(X)_{+}=\{f\in W_{+}^{(4)}(\tilde{X})(P^{(2)}(\tilde{X})_{+})|f\cdot D\geq 0\}.$

The subdiagram of $\Gamma(P(X)_{+})$ defined by all its black vertices coincides with $D$ . It is called Du Val part of $\Gamma(P(X)_{+})$ , and it is denoted by Duv $\Gamma(P(X)_{+})$ . Thus, Duv $\Gamma(P(X)_{+})=D\subset$ Duv

$\Gamma(P(\tilde{X})_{+})$

.

Double transparent vertices of $\Gamma(P(X)_{+})$ are identified with double trans(see Section 2.6), and single transparent verparent vertices of tices of $P(X)_{+}$ which are connected by two edges with double transpar. ent vertices of $\Gamma(P(X)_{+})$ are identified with such vertices of Indeed, they are orthogonal to the set which defines the reflection group as the group generated by reflections in all elements acts identically on all these ver. Thus, the group of tices, and all of them satisfy (66). All double transparent vertices and all single transparent vertices connected by two edges with double transparent vertices of $\Gamma(P(X)_{+})$ define the logarithmic part of $\Gamma(P(X)_{+})$ , and it is denoted by ${\rm Log}\Gamma(P(X)_{+})$ . Thus, we have $\Gamma(P(\tilde{X})_{+})$

$\Gamma(P(\tilde{X})_{+})$

$P^{(4)}(\tilde{X})_{+}$

$W_{+}^{(4)}(\tilde{X})$

$W_{+}^{(4)}(\tilde{X})$

$P^{(4)}(\tilde{X})_{+}$

${\rm Log}\Gamma(P(X)_{+})={\rm Log}\Gamma(P(\tilde{X})_{+})$

,

are identified. Moreover, the Du Val part logarithmic parts of $X$ and Duv $\Gamma(P(X)_{+})$ and the logarithmic part ${\rm Log}\Gamma(P(X)_{+})$ are disjoint in $\Gamma(P(X)_{+})$ because they are orthogonal to each other. Thus, the logarithmic with the same part of $\Gamma(P(X)_{+})$ is stable, it is the same for all pairs . On the Du Val part of $\Gamma(P(X)_{+})$ we have only a main invariants restriction: it is a subdiagram of Du Val part of one of extremal pairs ). described in Theorems 3.1 and 3.6 (with the same main invariants All vertices of $\Gamma(P(X)_{+})$ which do not belong to Duv $\Gamma(P(X)_{+})\cup$ ${\rm Log}\Gamma(P(X)_{+})$ define a subdiagram Var $\Gamma(P(X)_{+})$ which is called the $\tilde{X}$

$(X, \theta)$

$(r, a, \delta)$

$(X, \tilde{\theta})$

$(r, a, \delta)$

62

3. DPN SURFACES OF ELLIPTIC TYPE

varying part of $\Gamma(P(X)_{+})$ . By (66), we have Var $P(X)_{+}=\{f\in W_{+}^{(4)}(\tilde{X})(VarP(\tilde{X})_{+})|f\cdot D\geq 0\}$ skip when we consider only vertices). It describes Var $\Gamma(P(X)_{+})$ by the intersection pairing in and Of course, two Dynkin subdiagrams , with isomorphic Dynkin diagrams $D\cong D’$ , of two extremal with the same main invariants can give isomorphic and pairs Dynkin diagrams $\Gamma(P(X)_{+})$ and $\Gamma(P(X’)_{+})$ for defining by them K3 pairs . To have that, it is necessary and sufficient that root and invariants $([D],\xi^{+})$ and $([D’], (\xi’)^{+})$ defined by them are isomorphic. We remind that they can be obtained by restriction on $[D]$ and $[D’]$ of the root respectively, and they can be easily and invariants of pairs $([D],\xi^{+})$ and $([D’], (\xi’)^{+})$ isomorphic, computed. We remind that to have there must exist an isomorphism : $[D]\rightarrow[D’]$ of the root lattices and an of the discriminant quadratic form of the lattice automorphism . Section 2.5 gives the $veIy$ simple and effective which send for method for that. Thus, we have a very simple and effective method to find out when different subdiagrams $D$ above give K3 pairs with isomorphic diagrams. Note that we have used all equivalent conditions (i), (ii) and (iii) of Theorem 3.5 which shows their importance. Finally, we get (we

$\Gamma$

$S.$

$D\subset\Gamma(P^{(4)}(\tilde{X})_{+})$

$ D’\subset$

$\Gamma(P^{(4)}(\tilde{X}’)_{+})$

$(\tilde{X},\tilde{\theta})$

$(X, \theta)$

$(\tilde{X}’,\tilde{\theta’})$

$(X\prime, \theta’)$

$(\tilde{X}’,\tilde{\theta’})$

$(\tilde{X},\tilde{\theta})$

$\gamma$

$\overline{\phi}\in O(q_{S})$

$S$

$(\xi’)^{+}$

$\xi^{+}$

Theorem 3.7 (Classification Theorem in the non-extremal, i. e. arbitrary, case of elliptic type). Dynkin diagrams $\Gamma(P(X)_{+})$ of exceptional curves of non-extremal (i. e. arbitmry) non-symplectic involutions of elliptic $K3$ (without restrictions) Dynkin type of surfaces are described by arbitrary (see Theorem subdiagrams Duv of extremal pairs ). More(equivalently 3.6) with the same main invariants $(X, \theta)$

$ D\subset$

$\Gamma(P(\tilde{X})_{+})$

$(\tilde{X},\tilde{\theta})$

$(k, g, \delta)$

$(r, a, \delta)$

over,

Duv $\Gamma(P(X)_{+})=D,$

${\rm Log}\Gamma(P(X)_{+})={\rm Log}\Gamma(P(\tilde{X})_{+})$

,

and they are disjoint to each other,

Var $P(X)_{+}=\{f\in W_{+}^{(4)}(\tilde{X})(VarP(\tilde{X})_{+})|f\cdot D\geq 0\}$ is genemted by reflections in all elements of where the gmup Duv Duv Duv Dynkin subdiagrams (with the same main invariants) give $K3$ pairs with isomorphic Dynkin diagrams $\Gamma(P(X)_{+})\cong\Gamma(P(X’)_{+}),$ and only the $W_{+}^{(4)}(\tilde{X})$

$\Gamma(P(\tilde{X})_{+})=P^{(4)}(\tilde{X})_{+}.$

$ D\subset$

$\Gamma(P(\tilde{X})_{+}),$

$\Gamma(P(\tilde{X’})_{+})$

$ D’\subset$

$(X, \theta),$

$(X’, \theta’)$

$\iota f$

$\iota f$

3.3. CLASSIFICATION OF NON-SYMPLECTIC INVOLUTIONS

mot invariants $ D’\subset$

$([D], \xi^{+}),$

$([D’], (\xi’)^{+})$

Duv $\Gamma(P(X’)_{+})$ are isomorphic.

defined by

$ D\subset$

Duv

63 $\Gamma(P(\tilde{X})_{+})$

,

64

3. DPN SURFACES OF ELLIPTIC TYPE

TABLE 2. Diagrams $\Gamma(P(X)_{+})$ of extremal K3 surfaces of elliptic type which are different from Table 1 $(X, \theta)$

(In (a)

we repeat the corresponding case of Table

1)

3.3. CLASSIFICATION OF NON-SYMPLECTIC INVOLUTIONS

65

66

3. DPN SURFACES OF ELLIPTIC TYPE

3.3. CLASSIFICATION OF NON-SYMPLECTIC INVOLUTIONS

67

68

3. DPN SURFACES OF ELLIPTIC TYPE

3.4. Proof of Classification Theorem 3.6 Let be a non-symplectic involution of elliptic type of a K3 surface, , and is an extremal pair. with the main invariants $\Gamma(P(X)_{+})$ 3.5, is defined by the root subsystem By Theorem the where is a funcorresponding to . damental chamber of $W^{(2)}(S)$ , and $S=(S_{X})_{+}$ has the invariants where is a fundaWe can assume that of mental chamber of $W^{(2,4)}(S)$ defined by a choice of a basis (see Section 2.4.1). the root system be the Dynkin diagram of the root system Let the Weyl group of the root system and . We use the description of root subsystems given below. $(X, \theta)$

$(r, a, \delta)$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset\Delta^{(4)}(\mathcal{M}^{(2)})$

$(X, \theta)$

$(X, \theta)$

$\mathcal{M}^{(2)}$

$(r, a, \delta)$

$\mathcal{M}^{(2)}\supset \mathcal{M}(X)_{+}\supset \mathcal{M}^{(2,4)}$

$\mathcal{M}^{(2,4)}$

$P^{(4)}(\mathcal{M}^{(2,4)})$

$\Delta^{(4)}(\mathcal{M}^{(2)})$

$\Gamma(P^{(4)}(\mathcal{M}^{(2,4)}))$

$\triangle^{(4)}(\mathcal{M}^{(2)})$

$\triangle^{(4)}(\mathcal{M}^{(2)})$

$W^{(4)}(\mathcal{M}^{(2)})$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset$

$\triangle(4)(\mathcal{M}^{(2)})$

3.4.1 Let $T\subset R$ be a root subsystem of a root system $R$ and all components of $R$ have types $A,$ $D$ or $E$ . We consider two particular cases of root subsystems. Let $B$ be a basis of $R$ . Let $T\subset R$ be a primitive root subsystem. Then $T$ can be replaced by an equivalent root subsystem $\phi(T),$ $\phi\in W(R)$ , such

69

3.4. PROOF OF CLASSIFICATION THEOREM

that a part of the basis $B$ gives a basis of $T$ (see [Bou68]). Thus (up to equivalence defined by the Weyl group $W(R))$ , primitive root subsystems $T\subset R$ can be described by Dynkin subdiagrams . Now let $T\subset R$ be a root subsystem of a finite index. Let be a $R$ $j\in J$ , be a basis of . Let component of . Let be the maximal root of corresponding to this basis. Dynkin diagram of the set of roots $\Gamma\subset\Gamma(B)$

$R_{i}$

$R_{i}$

$r_{j},$

$r_{\max}=\sum_{j\in J}k_{j}r_{j}$

$R_{i}$

$\{r_{j}|j\in J\}\cup\{-r_{\max}\}$

is an extended Dynkin diagram of the Dynkin diagram $\Gamma(\{r_{j}|j\in J\})$ . Let us replace the component of the root system $R$ by the root subsystem having by its basis the set $(\{r_{j}|j\in J\}\cup\{-r_{\max}\})-\{r_{t}\}$ where $t\in J$ is some fixed element. We get a root subsystem $R’\subset R$ of finite index . It can be shown [Dyn57] that iterations of this procedure give any root subsystem of finite index of $R$ up to the action of $W(R)$ . Description of an arbitrary root subsystem $T\subset R$ can be reduced to these two particular cases, moreover it can be done in two ways. Firstly, any root subsystem $T\subset R$ is a subsystem of finite index where is a primitive root subsystem generated by $T.$ Secondly, any root subsystem $T\subset R$ can be considered as a primitive root subsystem where is root subsystem of finite index. $T$ One can take generated by and by any $u=$ rk $R-$ rk $T$ roots such that rk $[T, r_{1}, \ldots, r_{u}]=$ rk $R.$ $R_{i}$

$R_{i}’\subset R_{i}$

$k_{t}$

$ T\subset$

$T_{pr}$

$T_{pr}\subset R$

$R_{1}\subset R$

$T\subset R_{1}$

$R_{1}$

$r_{1},$

$\ldots,$

$r_{u}$

3.4.2 Here we show that the root subsystems which coincide with the . Obviously, full root systems can be realized by K3 pairs , and the Dynkin diagrams they are extremal. For them $\Gamma(P(X)_{+})=\Gamma(P(\mathcal{M}^{(2,4)}))$ coincide. All these diagrams are described in Table 1 of Theorem 3.1. It is natural to call such pairs superextremal. Thus, a non-symplectic involution of elliptic type of K3 (equivalently, the corresponding DPN pair$ (Y, C)$ or DPN surface) is called super-extremal, if for the corresponding root subsystem (equivalently, we have . We have $\triangle_{+}(\mathcal{M}^{(2)})$

$(X, \theta)$

$\triangle^{(4)}(\mathcal{M}^{(2)})$

$\mathcal{M}(X)_{+}=\mathcal{M}^{(2,4)}$

$(X, \theta)$

$(X, \theta)$

$\triangle_{+}^{(4)}(\mathcal{M}^{(2)})\subset$

$\triangle(4)(\mathcal{M}^{(2)})$

$\Delta_{+}^{(4)}(\mathcal{M}^{(2)})=\triangle^{(4)}(\mathcal{M}^{(2)})$

$\triangle_{+}^{(4)}=$

$\triangle^{(4)}(S))$

Proposition 3.8. For any possible elliptic triplet of main invariants there exists a super-extremal, $i.$

$e.$

$\Gamma(P(X)_{+})=\Gamma(P(\mathcal{M}^{(2,4)}))$

and standard (see Section 2. 7) $K3$ pair

$(X, \theta)$

.

,

$(r, a, \delta)$

70

3. DPN SURFACES OF ELLIPTIC TYPE

See the description of their graphs 1 of Theorem 3.1.

$\Gamma(P(X)_{+})=\Gamma(P(\mathcal{M}^{(2,4)}))$

in Table

and the Proof. Let us consider an elliptic triplet of main invariants which is described in Theocorresponding Dynkin diagram rem 3.1. Denote $K^{+}(2)=[P^{(4)}(\mathcal{M}^{(2,4)})]$ , i. e. it is the sublattice generated . Consider the corresponding root by all black vertices of invariant $(K^{+}(2), \xi^{+})$ , see (64) and (65). Consider $H=Ker\xi^{+}$ . By Propo, if the sitions 3.3 and 2.9, there exists a super-extremal standard pair inequalities $(r, a, \delta)$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$\Gamma(P(\mathcal{M}^{(2,4)}))$

$(X, \theta)$

$r+$

rk $K^{+}+l(\mathfrak{A}_{(K)_{p}}+)2,$

$r+a+2l(H)0$ and $(-K_{Z})$ $D>0$ for any curve $D$ on $Z$ . We have (see Section 1.5) $\sigma$

$\sigma(E)$

$\sigma$

$\Gamma(Y)$

${\rm Log}\Gamma(Y)$

$\sigma$

$Y\rightarrow Z$

$\sigma$

$\Gamma(Y)$

$\Gamma(Y)$

$\Gamma(Y)$

${\rm Log}\Gamma(Y)$

$\log$

$\leq 2$

$\sigma$

$\sigma$

${\rm Log}\Gamma(Y)$

$\pi$

$X\rightarrow Y$

$\theta$

$(X, \theta)$

$G_{i}\subset SL(2, \mathbb{C})$

$\mathbb{C}/G_{i}$

$(i.$

$e$

$\mathbb{C}/\tilde{G_{i}}$

$G_{i}\subset GL(2, \mathbb{C})$

$\overline{G_{i}}\cap SL(2, \mathbb{C})=G_{i}$

$\tilde{G_{i}}$

$\tilde{G_{i}}/G_{i}=\{1, \theta\}$

$\leq 2.$

$\log$

$4(-K_{Z})^{2}=(-2K_{Z})^{2}=(\sigma^{*}(-2K_{Z}))^{2}=(C_{g})^{2}>0.$

since $Y$ is a DPN surface of elliptic type. Moreover, $-2K_{Z}\cdot D=-2\sigma^{*}K_{Z}\cdot\sigma^{*}D=C_{g}\cdot\sigma^{*}D\geq 0$

is effective. Moreover, is irreducible with $(C_{g})^{2}>0$ and because consists of exceptional we get here zero, only if the effective divisor $F$ on $Y$ with $C_{g}\cdot F=0$ . But such curves $F$ correspond to vertices curves . They are contracted by into of the logarithmic or the Du Val part of points of $Z$ which is impossible for the divisor $\sigma^{*}D$

$C_{g}$

$\sigma^{*}D$

$\Gamma(Y)$

$\sigma$

$\sigma^{*}D.$

$\square $

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

106

del Pezzo surfaces of index Using Theorem 4.1, we can transfer to , the mot invariant, , equivalently the main invariants the mot subsystem, the exceptional curves which are defined for the surface $Y$ . In particular, the Picard number of of the right resolution : is ) $-\# V({\rm Log}\Gamma(Y))$ . (83) rk Pic $Z=r-\# V$ (Duv $Z$

$\log$

$\leq 2$

$(k, g, \delta)$

$(r, a, \delta)$

$\sigma$

$Z$

$Y\rightarrow Z$

$\tilde{r}=$

$\Gamma(Y)$

In Theorem 3.18 we have shown the Picard number in the extremal (for $Y)$ case. Obviously, surfaces with the extremal $Y$ are distinguished by (equivthe minimal Picard number for the fixed main invariants is prescribed by the main invaniants . Since the alently, . the numand is then fixed, this is equivalent to have the maximal rank $(-2)$ ber of -curves for the minimal resolution of singularities) for Du Val singularities of del Pezzo surfaces In Mori Theory, see [Mor82] and [Rei83], with rk Pic $Z=1$ are especially important. They give relatively minimal -tenminal singularities: any models in the class of rational surfaces with with -terminal singularities has a contracrational surface tion morphism onto such a model. From Theorems 4.1 and 3.6, we obtain -tenninal singularities of index classification of such models with By Theorem 3.18, they correspond to extremal DPN surfaces of elliptic type $\tilde{r}$

$Z$

$(r, a, \delta)$

$\tilde{r}$

$(k,g, \delta))$

${\rm Log}\Gamma(Y)$

$(i.$

$e$

$Z.$

$\log$

$Z$

$\log$

$X\neq \mathbb{P}^{1}\times \mathbb{P}^{1}$

$\log$

$\leq 2.$

$\log$

with

) $-\# V({\rm Log}\Gamma(Y))=1,$ (Duv and Theorem 3.18 gives the classification of the graphs of exceptional curves on them. This classification can be extended to a fine classification of the surfaces themselves. Here are results for the case of rk Pic $Z=1.$ $\tilde{r}=r-\# V$

$\Gamma(Y)$

Theorem 4.2. There exist, up to isomorphism, exactly 18 $logdel$ Pezzo surfaces $Z$ of index 2 with rk Pic $Z=1$ . The $DPN$ surfaces $Y$ of their right resolution ofsingularities are extremal and correspond to the following cases of Theorem 3.18, where we also show in parentheses the type of singularities of $Z$ : $11(K_{1}), 15(K_{1}A_{4}), 18(K_{1}A_{1}A_{5}), 19(K_{1}A_{7}), 20a(K_{1}D_{8})$ , $20b(K_{1}2A_{1}D_{6}), 20c(K_{1}A_{3}D_{5}), 20d(K_{1}2D_{4});21(K_{2}A_{2})$ , $25(2K_{1}A_{7}), 26(K_{2}2A_{3}), 27(K_{2}A_{7}), 30(K_{3}2A_{2});33(K_{3}A_{1}A_{5})$ ; $40(K_{5}), 44(K_{5}A_{4});46(K_{6}A_{2});50(K_{9})$

.

In particular, the isomorphism class of $Z$ is defined by its configuration of singularities. The number of non-Du Val singularities is at most one except when the singularities are $2K_{1}A_{7}.$ In all other cases 11-50 the surface with maximal Du Val part is also unique.

4.1. CLASSmCATION OF LOG DEL PEZZO SURFACES OF INDEX

$\leq 2$

107

Proof. For each of the graphs of Table 3 it is straightforward to pick a subgraph such that contracting the corresponding curves realizes $Y$ as a $n\leq 4$ . The images of the sequence of blowups starting from or $V=\mathbb{P}^{2}$

$\mathbb{F}_{n},$

remaining curves give a configuration of curves on By Theorem 3.20 we are guaranteed that, vice versa, starting with such a configuration, the corresponding series of blowups leads to a right resolution of singularities of a del Pezzo surface of index So, to compute the number of isomorphism classes, one has to find the orbits of the -action on the parameter space for the choices of the blowups. Finally, one has to take into account the action of the symmetry group of $V.$

$\tilde{Z}$

$Z$

$\log$

$\leq 2.$

$G$

the graph and the (finitely many) choices for the contractions to $V.$ In all the cases this is a straightforward computation which gives precisely one orbit. A typical case is that of case 48. The configuration of curves can be contracted to a mled surface so that the images of non-contracted curves are two distinct fibres, the exceptional section and an infinite section $s_{\infty}\sim s_{1}+f$ . In other words, they are the -invariant divisors on the toric variety . The blowups are uniquely determined except for the two last blowups corresponding to the two white end-vertices. One easily sees that these two blowups correspond to a choice of two points with on a toric surface lying on two toms orbits $Y’$ rk Pic $Y’=4$ . The surface corresponds to a polytope obtained from the polytope of by cutting two comers, which adds two new sides. These sides are obviously not parallel. Hence, the toms acts transitively on $Y$ , so the surface is unique. The only cases where a similar toric argument does not work are 39, 45 with 6 curves, and 47. In case 39 the surface $Y$ can be contracted to 3 sections and 3 fibres. This configuration is unique and the blowups are uniquely determined, so the surface $Y$ is unique. In case 45 the surface $Y$ , fibres is similarly contracted to with 6 curves, sections $C$ f$ $s\cap and $s’\cap f’$ and and curves $C\sim C’\sim s+f$ so that passes through $C’$ through $s\cap f’$ and $s’\cap f$ . This configuration is unique as well. with the followIn the most difficult case 47, $Y$ can be contracted to ing configuration: $\mathbb{F}_{1}$

$s_{1}$

$(\mathbb{C}^{*})^{2}$

$Y\rightarrow \mathbb{F}_{1}$

$\mathbb{F}_{1}$

$P_{1},$

$Y’\rightarrow \mathbb{F}_{1}$

$O_{1},$ $O_{2}$

$P_{2}$

$\mathbb{F}_{1}$

$(\mathbb{C}^{*})^{2}$

$O_{1}\times O_{2}$

$\mathbb{P}^{1}\times \mathbb{P}^{1}$

$\mathbb{P}^{1}\times \mathbb{P}^{1}$

$s,$

$s’$

$f,$ $f’$

$\mathbb{P}^{2}$

(1) three non-collinear points (2) three lines, each passing through two $P_{1},$

$l_{1},$

that

$l_{2},$

$P_{2},$

$P_{3},$

$l_{3}$

of the three points so

$P_{i}\not\in l_{i}.$

conics such that is tangent to lines and respectively at the points and ; and is tangent to lines and and respectively at the points

(3) two

$q_{1},$

$l_{2}$

$l_{3}$

$q_{1}$

$q_{2}$

$P_{3}$

$P_{2}$

$P_{3}$

$q_{2}$

$l_{1}$

$l_{3}$

$P_{1}.$

It is easy to see that this configuration is rigid as well.

$\square $

108

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

The Gorenstein case is well known” to experts but we were unable to find a complete and accurate description of the isomorphism classes in the literature. Therefore, we include the following theorem for completeness. Here we use the degree of a del Pezzo surface $Z$ which is $d=K_{Z}^{2}.$ $d$

There exist $28$ configurations ofsingularities of GorenTheorem 4.3. stein $log$ del Pezzo surfaces of Picard number 1, and each type determines the corresponding surface up to a deformation. The types (and the cases $N$ in Table 3) are as follows: $(a)$

(1) $ d=9:\emptyset$ (case1) (2) $d=8:A_{1}(2)$ (3) $d=6:A_{2}A_{1}(5)$ (4) $d=5:A_{4}(6)$ (5) $d=4:D_{5}(7a)A_{3}2A_{1}(7b)$

(6) $d=3:E_{6}(8a),$ $A_{5}A_{1}(8b),$ $3A_{2}(8c)$ (7) $d=$ 2: $E_{7}(9a),$ $A_{7}(9b),$ $A_{5}A_{2}(9c),$

$2A_{3}A_{1}(9d),$

$D_{6}A_{1}(9e)$

,

$D_{4}3A_{1}(9f)$

(8) $d=1:E_{8}(10a),$

$A_{8}(10b),$ $A_{7}A_{1}(10c),$ $A_{5}A_{2}A_{1}(10d),$

$2A_{4}(10e)$ ,

$D_{8}(10fl, D_{5}A_{3}(10g) , E_{6}A_{2}(10h) , E_{7}A_{1}(10i) , D_{6}2A_{1}(10j)$ ,

. In each type there is exactly one isomorphism class, with the folthere are two isomorphism lowing exceptions: in types classes; and in type there are infinitely many isomorphism classes parameterized by and all surfaces of The three extra surfaces of type are distinguished by the fact that their automorphism gmups are type 2 1-dimensional and contain . All other surfaces with $d=1$ have finite automorphism gmups. $2 D_{4}(10k), 2A_{3}2A_{1}(10l), 4A_{2}(10m)$

$(b)$

$E_{8},$

$E_{7}A_{1},$

$E_{6}A_{2}$

$2D_{4}$

$A^{1}.$

$E_{8},$

$(c)$

$E_{7}A_{1},$

$E_{6}A_{2}$

$D_{4}$

$\mathbb{C}^{*}$

lst proof. The first case to consider is $d=1$ . Choosing an appropriate subgraph in the graph of exceptional curves on , one picks a sequence of . These contractions and images of $(-2)$ -curves are listed blowups in [BBD84]. In addition, one has to compute the images of $(-1)$ -curves. The result is a configuration of lines, conics and cubics on , and in most $\tilde{Z}$

$\tilde{Z}\rightarrow \mathbb{P}^{2}$

$\mathbb{P}^{2}$

cases cubics can be avoided. Again, by theorem 3.20 we are guaranteed that, vice versa, starting with such a configuration, the conesponding series of blowups leads to a minimal del Pezzo surface resolution of singularities of a Gorenstein To compute the automorphism groups and the number of isomorphism PGL(3) of a projective classes, one has to compute the stabilizer and the orbits of the -action on the parameter space configuration on $\tilde{Z}$

$\log$

$Z.$

$ G\subset$

$\mathbb{P}^{2}$

$G$

for the configurations and the choices for the blowups; and to take into

4.1. CLASSIFICATION OF LOG DEL PEZZO SURFACES OF INDEX

109

$\leq 2$

account the action of the symmetry group of the graph and the (finitely many) choices for the contractions to In the case , the projective configuration is a line and a point on it, the group $G$ is the subgroup of upper-triangular matrices, and the parameter space is which can be identified with the set of power series $\mathbb{P}^{2}.$

$E_{8}$

$\mathbb{C}^{*}\times \mathbb{C}^{4}$

$y=\alpha_{3}x^{3}+\alpha_{4}x^{4}+\alpha_{5}x^{5}+\alpha_{6}x^{6}+\alpha_{7}x^{7}$

$mod x^{8}$

with

$\alpha_{3}\neq 0.$

The $G$ -action has two orbits: those of $y=x^{3}$ and of $y=x^{3}+x^{7}$ . The first orbit is in the closure of the second. The stabilizer of $y=x^{3}$ is isomorphic , the second stabilizer is to and consists of diagonal matrices finite. The model for the moduli stack is with -action $(1, c, c^{3})$

$\mathbb{C}^{*}$

$[A^{1} :

$\lambda.a=$

$\mathbb{C}^{*}$

\mathbb{G}_{m}]$

$\lambda^{4}a.$

, the projective configuration is a line , a conic In the case tangent to it, and another line . There are two cases: when intersects at 2 distinct points, and when they are tangent. One case is a degeneration of another, and in the degenerate case the stabilizer of the configuration contains , the projective configuration consists of 4lines and In the case 3 of them either pass through the same point or they do not. Once again, the local model is with the standard action, one configuration degenerates into another, and the degenerate configuration has stabilizer , the projective configuration consists of 4lines through a In the case $P$ and . The parameter space for such configuration point the 5th line ( points). Dividing by the symmetry group is gives as the stabilizer group. Every configuration has In all other cases for $d=1$ the computation gives one isomorphism class. For $d=2$ , the surfaces are obtained from the surfaces of $d=1$ by contracting one $(-1)$ -curve. So, the cases where more than one isomorphism class is possible are the ones that come from the four exceptional cases above. The only contraction of the -case is the case . In this case, the group of upper triangular matrices acts on the polynomials $y=\alpha_{3}x^{3}+\cdots+$ $\alpha_{6}x^{6}mod x^{7}$ with transitively; so there is only one isomorphism class. . In each of these, the surface and produces The case is unique because it can also be obtained by contracting a surface of type and , respectively. hich also comes from the type The case produces produces with a unique isomorphism class. Similarly, the case . For $d\geq 3$ , moreover, there is only which also comes from the type one isomorphism class for each configuration of singularities. $E_{7}A_{1}$

$l_{1}$

$l_{2}$

$l_{2}$

$q$

$q$

$\mathbb{C}^{*}$

$E_{6}A_{2}$

$[A^{1} :

\mathbb{G}_{m}]$

$\mathbb{C}^{*}$

$2D_{4}$

$l_{5}\geq P$

$\mathbb{P}^{1}\backslash $

$3$

$\mathbb{Z}/2\times \mathbb{Z}/2$

$A^{1}$

$\mathbb{C}^{*}$

$\tilde{Z}_{1}$

$\tilde{Z}_{2}$

$E_{7}$

$E_{8}$

$\alpha_{3}\neq 0$

$E_{7}A_{1}$

$D_{6}2A_{1}$

$D_{6}A_{1}$

$E_{7}$

$E_{8}$

$E_{6}A_{2}$

$A_{5}A_{1},$ $w$

$A_{5}A_{2}A_{1}$

$2D_{4}$

$D_{4}3A_{1},$

$D_{6}2A_{1}$

$\square $

110

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

2nd proof for the $d=1$ case. By Theorem 1.5 and Remark 1.7, a general element of the linear system $|-K_{Z}|$ is smooth. By Riemann-Roch theorem, $h^{0}(-K_{Z})=2$ . Hence, $|-K_{Z}|$ is a pencil with a unique, nonsingular base point $P$ . The blowup of $Z$ at $P$ is an elliptic surface with a fibration : and a section. The condition rk Pic $Z=1$ implies that the is an extremal mtional elliptic minimal resolution of singularities surface, as defined in [MP86]. Vice versa, given an extremal relatively minimal (with no $(-1)$ -curves with a section, contracting the $(-2)$ -curves not in fibres of ) surface meeting the section and then the section gives a Gorenstein del Pezzo surface with Du Val singulanities and rk Pic $Z=1$ . The finitely many choices of a section differ by the action of the Mordell-Weil group of the elliptic fibration, and hence give isomorphic $Z’ s.$ Hence, the classification of Gorenstein del Pezzo surfaces of degree 1 and rank 1 is equivalent to the classification of extremal rational elliptic fibrations with a section. The latter was done by Miranda and Persson in [MP86], and we just need to translate it to del Pezzo surfaces. On the level of graphs of exceptional curves, the transition from to consists of inserting an extra $(-1)$ -curve and changing $(-1)$ -curves through $P$ to $(-2)$ -curves. The tum into the corresponding ex. In addition, and can tum into tended Dynkin graphs are not seen respectively. The elliptic fibres $Z.$ in the graphs of of elliptic fibrations. According to [MP86, Thm 4.1] there are 16 are distinguished by and The four special is constant and the fact that the induced modular -function : there are exactly two singular fibres. The subgroup $Aut_{j}Y$ of automorphisms commuting with is always finite. Hence, in the four exceptional cases Aut $Y$ has dimension one and ective, and hence the automorIn all other cases -map is contains phism group is finite. By [MP86, Thm 5.4], in fifteen of the sixteen cases the elliptic surface , there are infinitely many isomolphism classes, is unique. In the case one for each value $\pi$

$Z’\rightarrow \mathbb{P}^{1}$

$\tilde{Z}’\rightarrow \mathbb{P}^{1}$

$\tilde{Z}’$

$\pi$

$\log$

$\tilde{Z}$

$\tilde{Z}’$

$gp_{\sim}$

$\tilde{A}_{n},$

$graphs*\tilde{A}_{1},$

$D_{n},\tilde{E}_{n}$

$A_{1}$

$A_{2}$

$\tilde{A}_{0}and*\tilde{A}_{0}$

$*\tilde{A}_{2}$

$tyP_{\sim}^{es}$

$types*\tilde{A}_{0}\tilde{E}_{8},$

$*\tilde{A}_{1}\tilde{E}_{7},$

$*\tilde{A}_{2}\tilde{E}_{6}$

$j$

$2D_{4}$

$j$

$\mathbb{P}^{1}\rightarrow \mathbb{P}_{j}^{1}$

$j$

$\mathbb{C}^{*}$

$j$

$su\dot{\eta}$

$2\tilde{D}_{4}$

$\square $

$j\in A_{j}^{1}.$

and without Du If we consider del Pezzo surfaces $Z$ of index one, every singuVal singularities, we get an opposite case to the previous $Z$ larity of must have index 2. This case includes and is sulprisingly similar to the classical case of non-singular del Pezzo surfaces when there are no singularities at all. Applying The\’orem 4.1 we get the following $\log$

$\leq 2$

4.1. CLASSIFICATION OF LOG DEL PEZZO SURFACES OF INDEX

111

$\leq 2$

Theorem 4.4. Up to deformation, there exist exactly 50 types of $logdel$ Pezzo surfaces $Z$ with singularities of index exactly 2 a singularity does exist). The $DPN$ surfaces $Y$ of their right resolution of singularities have $zem$ mot invariants, and are defined by empty $DuVal$ part Duv $(equivalently (k, g, \delta)$), up to their main invariants deformation (the can be obtained moduli are irreducible and connected). The diagram (with the same main invarifrom the diagram of cases 1–50 of Table 3 ants) as follows: and consists of $(lf$

$\Gamma(Y),$

$(r, a, \delta)$

$\Gamma(Y)$

$\Gamma$

${\rm Log}\Gamma(Y)={\rm Log}\Gamma$

$\Gamma(Y)$

Var $\Gamma(Y)=W$ (Var )

(84)

$\Gamma$

. one where $W$ is generated by reflections in all vertices of Duv should take in Theorem 3.19). In cases 7, 8, 9, 10, 20 one can $8a,$ $9a,$ $10a$ and $20a$ (diagrams consider only diagrams of cases $9\wedge f10b-m,$ $20b-d$ give the same). The type of Dynkin diagram Duv can be considered as analogous to the type of mot system which one usually associates to non-singular $del$ Pezzo surfaces. Its actual meaning is to give the type of the $Wyl$ gmup $W$ describing the varying part Var by (84). In cases 7–10, 20, one should (or can) take graphs of cases $7a-10a,$ $20a.$ $\Gamma(i.$

$e$

$ D=\emptyset$

$7b,$

$7a,$

$\Gamma$

$8b_{J}c,$

$\Gamma$

$(\Gamma(Y))$

$\Gamma$

Proof. This case corresponds to $Y$ with empty Duv of Theorem 9, 8, 7, Thus, 10 or 20 give the 3.19. Then the root invariant is . all cases we can consider only isomorphic root invariants and the same diagrams, and the corresponding cases $7a,$ $8a,$ $9a,$ $10a$ and $20a$ to calculate the diagrams. Let us show that moduli spaces of DPN surfaces $Y$ with the same main invariants and zero root invariant $(i. e. D=0)$ are irreducible. It is enough to show irreducibility of the moduli space of the corresponding right DPN pairs $(Y, C)$ where $C\in|-2K_{Y}|$ is non-singular. Taking double covering : $X\rightarrow Y$ ramified in $C$ , it is enough to prove irreducibility of moduli Mod of K3 surfaces with non-symplectic in$(S_{X})_{+}=S$ . General volutions and where has invariants invariant, want, have such pairs have zero root because general as we $S_{X}=S$ . Irreducibility of Mod had been discussed in Section 2.3 with proofs in Appendix: Section A.2. $ D\subset$

$\Gamma$

$0$

$(r, a, \delta)$

$\pi$

$(r,a,\delta)$

$S$

$(X, \theta)$

$(r, a, \delta)$

$(X, \theta)$

$(r,a,\delta)$

$\square $

We remark that the result equivalent to Theorem 4.4 was first obtained

in [Nik83]. We remark that cases 1–10 of Theorem 4.4 give classical non-singular del Pezzo surfaces. Therefore, Theorem 4.4 and all results of this work are very similar to classical show that del Pezzo surfaces of index non-singular del Pezzo surfaces. $\log$

$\leq 2$

112

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

4.2. Example: Enumeration of all possible types for $N=20$ Let us consider enumeration of all types of singularities and graphs of exdel Pezzo surfaces of index 2 of type $N=20$ , i. e. ceptional curves of with the main invariants $(r, a, \delta)=(10,8,1)$ . From Theorem 4.1 and Table 3, cases $20a-d$ , we obtain that all of them have one singulanity of index 2 and Du Val singulanities which $D_{6}2A_{1},$ and 2 . It correspond to a subgraph of one of graphs follows that their Du Val singularities are exactly of one of 52 types listed below: $\log$

$K_{1}$

$D_{8},$

$2D_{4}$ $D_{8}$

$D_{7}$

;

$D_{6}$

;

$D_{5}A_{3},$ $D_{5}A_{2},$ $D_{5}2A_{1},$ $D_{5}A_{1},$ $D_{5}$

;

$D_{4}A_{3},$ $D_{4}A_{2},$ $D_{4}3A_{1},$ $D_{4}2A_{1},$ $D_{4}A_{1},$ $D_{4}$

$A_{6}$

;

; ;

$A_{5}2A_{1},$ $A_{5}A_{1},$

$A_{5}$

;

$A_{4}A_{3},$ $A_{4}A_{2},$ $A_{4}2A_{1},$ $A_{4}A_{1},$

2

$D_{4}$

; ;

$D_{6}2A_{1},$ $D_{6}A_{1},$

$A_{7}$

$D_{5}A_{3}$

$A_{4}$

;

$A_{3}A_{1},2A_{3},$ $A_{3}A_{2}2A_{1},$ $A_{3}A_{2}A_{1},$ $A_{3}A_{2},$ $A_{3}4A_{1},$ $A_{3}3A_{1},$ $A_{3}2A_{1},$ $A_{3}A_{1},$

$2A_{2}2A_{1},2A_{2}A_{1},2A_{2},$ $A_{2}4A_{1},$ $A_{2}3A_{1},$ $A_{2}2A_{1},$

$A_{2}A_{1},$

$A_{2}$

$A_{3}$

;

;

$6A_{1},5A_{1},4A_{1},3A_{1};2A_{1},$

$A_{1};\emptyset.$

Using calculations of root invariants of Lemma 3.12, it is easy to calculate the root invariant for any of the subgraphs. One can see that it is defined uniquely by the type of Du Val singularities except the following 15 types of Du Val parts of singularities for which we show all differences in their root invariants. : There are exactly two possibilities for the root invariant (and then for the dual graph of exceptional curves). The first one can be obtained (case $20a$ ), and the second by taking by taking as a subdiagram in (case $20c$ ). In the second case, the charas a subdiagram in acteristic element can be written using elements of the component , and this is impossible in the first case. : There are exactly two possibilities for the root invariant. The (case $20a$ ), as a subdiagram in first one can be obtained by taking (case $20b$ ). and the second one by taking as a subdiagram in In the second case, the characteristic element can be written using elements , and it is impossible in the first case. of the components $D_{4}A_{3}$

$D_{4}A_{3}$

$D_{8}$

$D_{5}A_{3}$

$D_{4}A_{3}$

$A_{3}$

$D_{4}2A_{1}$

$D_{4}2A_{1}$

$D_{4}2A_{1}$

$2A_{1}$

$D_{8}$

$D_{6}2A_{1}$

113

4.2. ALL POSSIBLE TYPES FOR $N=20$

: There are exactly two possibilities for the root invariant. The group $H=\{0\}$ or . Both cases can be obtained by taking subdiagrams $A_{7}$

$H\cong \mathbb{Z}/2$

in

(case

$D_{8}$

$A_{5}A_{1}$

$20a$ ).

: There are exactly two possibilities for the root invaniant: the . Both cases can be obtained by taking subdiaor

group (case $20a$ ). grams in : There are exactly four possibilities for the root invariant. For the group $H=\{0\}$ the characteristic element can be wlitten using elements either of one component or only by both components . For the group either $\alpha=1$ or $\alpha=0$ . Three of these cases can be obtained by (case $20a$ ). The remaining case taking subdiagrams in and $\alpha=0$ (case $20c$ ). can be obtained by taking a subdiagram in : There are exactly two possibilities for the root invariant: $\alpha=0$ $\alpha=1$ (case or . Both cases can be obtained by taking subdiagrams in $20a)$ . : There are exactly two possibilities for the root invaniant: In the first case the characteristic element cannot be written using elements of the (it can be obtained by taking a subdiagram in , i. e. for components $20a$ ). the case For the second case it can be written using elements of the (it can be obtained by taking a subdiagram in component for the case $20b$ ). : There are exactlyfive possibilities for the root invariant. For the group $H=\{0\}$ the characteristic element can be written using elements ei, or using all three compother of one component , or by components nents . For the group either or $\alpha=0$ . Four of these (case $20a$ ). The remaincases can be obtained by taking subdiagrams in ing case and $\alpha=0$ can be obtained considering a subdiagram in $H=\{0\}$

$H\cong \mathbb{Z}/2$

$D_{8}$

$2A_{3}$

$A_{3}$

$A_{3}$

$H\cong \mathbb{Z}/2$

$H\cong \mathbb{Z}/2$

$D_{8}$

$D_{5}A_{3}$

$A_{3}A_{2}$

$D_{8}$

$A_{3}3A_{1}$

$A_{3}$

$D_{8}$

$D_{6}2A_{1},$

$A_{3}$

$i.$

$e.$

$A_{3}2A_{1}$

$2A_{1}$

$A_{3}$

$A_{3}2A_{1}$

$\alpha=1$

$H\cong \mathbb{Z}/2$

$D_{8}$

$H\cong \mathbb{Z}/2$

$D_{6}2A_{1}$

(case

$20b$ ).

: There are exactly two possibilities for the root invariant: $\alpha=1$ . Both cases can be obtained by taking subdiagrams in or $A_{3}A_{1}$

$\alpha=0$

$D_{8}$

$20a)$

(case

.

: There are exactly two possibilities for the root invariant: $\alpha=0$ or . Both cases can be obtained by taking subdiagrams in (case $20a$). : There are exactly two possibilities for the root invaniant: $\alpha=1$ (case . Both cases can be obtained by taking subdiagrams in or $A_{3}$

$\alpha=1$

$D_{8}$

$\alpha=0$

$A_{2}2A_{1}$

$D_{8}$

$20a)$

.

: There are exactly two possibilities for the root invariant. For the first one the characteristic element can be written using two pairs of compo. For the second one the characteristic element can be written nents of using only one pair of components of : There are exactly four possibilities for the root invariant. For the group $H=\{0\}$ the characteristic element can be written using elements $5A_{1}$

$5A_{1}$

$5A_{1}.$

$4A_{1}$

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

114

or by only four components . For the group either of two components either $\alpha=1$ or $\alpha=0$ . All four cases can be obtained by taking (case $20c$ ). subdiagrams in 2 : There are exactly two possibilities for the root invariant: $\alpha=0$ or $\alpha=1$ . (case Both cases can be obtained by considering subdiagrams in $A_{1}$

$A_{1}$

$H\cong \mathbb{Z}/2$

$D_{4}$

$3A_{1}$

$D_{8}$

$20a)$

. : There are exactly two possibilities for the root invaniant: . Both cases can be obtained by considering subdiagrams in

$2A_{1}$

$\alpha=1$

$20a)$

$\alpha=0$ $D_{8}$

or

(case

.

Thus, for the types of Du Val singularities shown above. (together with

of index two) we obtain the number shown above of difdel Pezzo surfaces: their right resolution of singulanities can have that number of different graphs of exceptional curves. By taking the corresponding sequence of contractions of-l curves, one can further investigate these surfaces in details; in particular, one can enumerate irre-

the singulanity ferent types of

$K_{1}$

$\log$

ducible components of their moduli. Thus, there are exactly $52+12+3\cdot 2+4=74$ different graphs of exceptional curves on the right resolution of singularities of log del Pezzo surfaces of index 2 with the main invariants $(r, a, \delta)=(10,8,1)(i.$ $e.$

$N=20)$ .

Of course, similar calculations can be done for all 50 types of main . The considered case del Pezzo surfaces of index invariants of $N=20$ is one of the richest and most complicated. $\leq 2$

$\log$

4.3. Application: Minimal projective compactifications of del by relatively minimal affine surfaces in $\mathbb{P}^{2}$

$\log$

Pezzo surfaces of index

$\leq 2.$

This is similar to [BBD84] in the Gorenstein case. Let us consider one of the 45 relatively minimal surfaces of Theorems ( 4.2, 4.3 which are different from . except the case 1). Let : a sequence be its right resolution of singularities, and such that the corresponding exceptional curves on $Y$ of vertices of give a contraction of the sequence of curves of the lst kind : Let be the union of images by of all exceptional curves $v\in V$ (Duv $(\Gamma(Y))$ ) $UV({\rm Log}(\Gamma(Y)))$ . Then, the embedding $f=(\sigma\tau^{-1})$ : gives a compactification of the affine surface $W=\mathbb{P}^{2}-C$ of The morphism is minimal in the sense that cannot be extended through components of $C$ (see [BBD84] for details). The description of all such affine surfaces $W$ and such their compactifications is then reduced to the $\mathbb{P}^{2}$

$i.$

$e$

$\sigma$

$Y\rightarrow Z$

$v_{1},$

$\ldots,$

$v_{r-1}$

$\Gamma(Y)$

$Y\rightarrow \mathbb{P}^{2}.$

$\tau$

$C\subset \mathbb{P}^{2}$

$E_{v},$

$\tau$

$W\rightarrow Z$

$\mathbb{P}^{2}.$

$f$

$f$

115

4.4. DIMENSION OF THE MODULI SPACE

$(defined by v_{1}, \ldots, v_{r-1}))$ which were description of subdiagrams of described by their connected components in Section 3.6. $\Gamma(Y)$

4.4. Dimension of the moduli space For each triple of invariants (85)

$(k, g, \delta, the root invariant)$

,

equivalently, (86)

$(r, a, \delta, the dual diagram of exceptional curves \Gamma(Y)$ )

one has the moduli space of pairs of log del Pezzo surfaces together $C\in|-2K_{Z}|$ with a smooth curve . We have established the equivalence C)$ $(Z, with a non-symplectic invoK3 between pairs and surfaces lution. Hence instead of moduli of pairs $(Z, C)$ we can consider . moduli of pairs $\mathcal{M}_{(Z,C)}$

$(X, \theta)$

$\mathcal{M}_{(Z,C)}$

$\mathcal{M}_{(X,\theta)}$

$(X, \theta)$

By (63), (87)

$\dim \mathcal{M}_{(X,\theta)}=20-r-\# V(Duv(\Gamma))=9+g-k-\# V(Duv(\Gamma))$

.

Moreover, (88)

$\dim|-2K_{Z}|=\dim|C_{g}|=3g-3$

Sections 1.4 and 1.5). It follows that the dimension of the parameter space and with the graph ofgeneric surfaces of type $Y$ of exceptional curves on the right resolution of singularities (or with the corresponding root invariant) is equal to (see

$Z$

$\mathcal{M}_{(r,a,\delta),\Gamma(Y)}$

$(r, a, \delta)$

$\Gamma(Y)$

$\dim \mathcal{M}_{(r,a,\delta),\Gamma(Y)}=12-2g-k-\# V($

Duv $(\Gamma(Y))+\dim$ Aut $Z=$

(89) $\frac{r+3a}{2}-10-\# V(Duv(\Gamma(Y))+\dim$

Aut $Z$

Note that this formula may fail for non-generic surfaces. For example, by Theorem 4.3 there are exactly two isomorphism classes of Gorenstein surfaces with a single -singularity. The formula above gives $E_{8}$

$\dim \mathcal{M}_{(r,a,\delta),\Gamma(Y)}=\dim$

Aut $Z$

which is tme for the generic surface that has trivial isomorphism group and fails for the second surface which has Aut $Z=\mathbb{C}^{*}$

116

4. CLASSIFICATION OF LOG DEL PEZZO SURFACES

4.5. Some open questions 4.5.1. Finite characteristic It would be very interesting to generalize results of this work to finite characteristic. As we had mentioned in Remark 3.16, it seems, the main problem is to generalize Theorem 1.5. We think that our results are valid in . As we have seen, in characteristic 2 the number of cases characteristic $\geq 3$

increases.

4.5.2. Arithmetic of log del Pezzo surfaces of index

$\leq 2$

There are many results (e.g. see [Man86], [MT86] and [CT88]) where the anithmetic of classical non-singular del Pezzo surfaces is studied. What is del Pezzo surfaces of index and equivalent DPN the anithmetic of surfaces of elliptic type? $\log$

$\leq 2$

Appendix

Here we add some important results and calculations which were used in the main part of the work and which are important by themselves.

A.l. Integral symmetric bilinear forms. Elements of the discriminant forms technique we review results about integral symmetric bilinear forms (lattices) which we used. We follow $[Nik80b].$ Here, for readers’ convenience,

A.l.1. Lattices

Everywhere in the sequel, by a lattice we mean a free -module of finite rank, with a nondegenerate symmetric bilinear form with values in the ring of rational integers (thus, “lattice” replaces the phrase “nondegenerate integral symmetric bilinear form”). A lattice $M$ is called even if $x^{2}=x\cdot x$ is even for each $x\in M$ , and odd the value of the bilinear form of $M$ at the otherwise (here we denote by we denote the orthogonal direct sum of lattices pair $(x, y))$ . By . If $M$ is a lattice, we denote by $M(a)$ the lattice obtained from and $M$ by multiplying the form of $M$ by the rational number $a\neq 0$ , assuming that $M(a)$ is also integral. $\mathbb{Z}$

$\mathbb{Z}$

$x\cdot y$

$M_{1}\oplus M_{2}$

$M_{2}$

$M_{1}$

A.1.2. Finite symmetric bilinear and quadratic forms By a finite symmetric bilinear form we mean a symmetric bilinear form defined on a finite Abelian group : satisfying By a finite quadratic form we mean a map : the following conditions: and 1 $q(na)=n^{2}q(a)$ for all $b$

$\mathfrak{A}\times \mathfrak{A}\rightarrow \mathbb{Q}/\mathbb{Z}$

$\mathfrak{U}.$

$q$

$)$

$n\in \mathbb{Z}$

$a\in \mathfrak{U}.$

117

$\mathfrak{U}\rightarrow \mathbb{Q}/2\mathbb{Z}$

118

APPENDIX

2 $\mathbb{Q}/\mathbb{Z}$

of

where is a finite symmetric bilinear fonn, which we call the bilinear form $q(a+a’)-q(a)-q(a’)\equiv 2b(a, a’)(mod 2)$ ,

$)$

$ b:\mathfrak{U}\times \mathfrak{A}\rightarrow$

$q.$

A finite quadratic form is nondegenerate when is nondegenerate. In and of orthogthe usual way, we introduce the notion of orthogonality onal sum of finite symmetric bilinear and quadratic forms. $b$

$q$

$(\perp)$

$(\oplus)$

A.1.3. The discriminant form of a lattice The bilinear form of a lattice $M$ determines a canonical embedding $M^{*}=Hom(M, \mathbb{Z})$ . The discriminant group of the lattice $M$ is the factor group $\mathfrak{A}_{M}=M^{*}/M$ . It is finite and Abelian, and its order is equal to $|\det(M)|$ . We remind that the determinant $\det(M)$ of $M$ equals for some basis of the lattice $M.$ $A$ lattice is called unimodular if $ M\subset$

$\det(e_{i}\cdot e_{j})$

$L$

$e_{i}$

$\det(L)=\pm 1.$

We extend the bilinear fonn of $M$ to one on

$M^{*}$

,

taking values in

$\mathbb{Q}$

. We

put $b_{M}(t_{1}+M, t_{2}+M)=t_{1}\cdot t_{2}+\mathbb{Z}$

where

$t_{1},$

$t_{2}\in M^{*}$

, and $q_{M}(t+M)=t^{2}+2\mathbb{Z},$

if $M$ is even, where $t\in M^{*}$ : and We obtain the discriminant bilinear form $M$ (if : is even) of the the discriminant quadratic form $M$ lattice . They are nondegenerate. Similarly, one can define discriminant forms of -adic lattices over the as a sum ring of -adic integers for a prime . The decomposition of and as the defines the decomposition of of its -components . They are equal to the and orthogonal sum of its -components discriminant forms of the corresponding -adic lattices We denote by: the 1-dimensional -adic lattice detennined by the matrix ); (taken where $k\geq 1$ and $b_{M}$

$q_{M}$

$\mathfrak{A}_{M}\times \mathfrak{U}_{M}\rightarrow \mathbb{Q}/\mathbb{Z}$

$\mathfrak{U}_{M}\rightarrow \mathbb{Q}/2\mathbb{Z}$

$p$

$\mathbb{Z}_{p}$

$\mathfrak{U}_{M}$

$p$

$p$

$b_{M}$

$(\mathfrak{A}_{M})_{p}$

$p$

$(b_{M})_{p}$

$p$

$(q_{M})_{p}$

$p$

$K_{\theta}^{(p)}(p^{k})$

$U^{(2)}(2^{k})$

$M\otimes \mathbb{Z}_{p}.$

$\langle\theta p^{k}\rangle,$

$p$

$\theta\in \mathbb{Z}_{p}^{*}$

and

$V^{(2)}(2^{k})$

$q_{M}$

$mod (\mathbb{Z}_{p}^{*})^{2}$

the 2-dimensional2-adic lattices determined by the

matrices $\left(\begin{array}{ll}0 & 2^{k}\\2^{k} & 0\end{array}\right), \left(\begin{array}{ll}2^{k+l} & 2^{k}\\2^{k} & 2^{k+1}\end{array}\right)$

respectively; $q_{\theta}^{(p)}(p^{k}),$

$U^{(2)}(2^{k})$

$u_{+}^{(2)}(2^{k})$

and

and

$v_{+}^{(2)}(2^{k})$

$V^{(2)}(2^{k})$

, the discriminant quadratic forms of

respectively;

$K_{\theta}^{(p)}(p^{k})$

,

119

A. 1. DISCRIMINANT FORMS TECHNIQUE $b_{\theta}^{(p)}(p^{k}),$

$v_{+}^{(2)}(2^{k})$

and respectively. $u_{-}^{(2\rangle}(2^{k})$

$v_{-}^{(2)}(2^{k})$

, the bilinear forms of

$q_{\theta}^{(p)}(p^{k}),$

and

$u_{+}^{(2)}(2^{k})$

These -adic lattices and finite quadratic and bilinear forms are called elementary. Any -adic lattice (respectively finite non-degenerate quadratic, bilinear form) is an orthogonal sum of elementary ones (Jordan decompo$p$

$p$

sition).

A.1.4. Existence of an even lattice with a given discriminant quadratic form The signature of a lattice $M$ is equal to $signM=t_{(+)}-t_{(-)}$ where and are numbers of positive and negative squares of the corresponding real form . The formula $t_{(+)}$

$t_{(-)}$

$M\otimes \mathbb{R}$

$signq_{M} mod 8=signM mod 8=t_{(+)}-t_{(-)} mod 8$

where $M$ is an even lattice, correctly defines the signature $mod 8$ for nondegenerate finite quadratic forms. For elementary finite quadratic forms we obtain respectively $signq_{\theta}^{(p)}(p^{k})\equiv k^{2}(1-p)+4k\eta mod 8$

where

$p$

is odd and

$(\frac{\theta}{p})=(-1)^{\eta}$

,

where we use Legendre symbol;

$signq_{\theta}^{(2)}(2^{k})\equiv\theta+4\omega(\theta)k mod 8$

where $\omega(\theta)\equiv(\theta^{2}-1)/8mod 2$ ; $signv_{+}^{(2)}(2^{k})\equiv 4k mod 8$

;

$signu_{+}^{(2)}(2^{k})\equiv 0 mod 8.$

In particular, $signL\equiv$ Omod8 if is an even unimodular lattice. We denote by the minimal number of generators of a finite Abelian group . We consider an even lattice $M$ with the invariants $(t_{(+)}, t_{(-)}, q)$ where are its numbers of positive and negative squares, and ; we denote by the group where is defined. The invariants q are equivalent to the genus of $M$ (see Corollary 1.9.4 in $[Nik80b]$ ) Thus, they define the isomorphism classes of the -adic lattices for all prime , and of We have (see Theorem 1. 10. 1 in $[Nik80b]$ ): $L$

$l(\mathfrak{A})$

$\mathfrak{U}$

$t_{(+)},$

$q_{M}$

$ q\cong$

$t_{(-)}$

$\mathfrak{U}_{q}$

$(t_{(+)},$ $t_{(-)},$

$q$

$)$

$M\otimes \mathbb{Z}_{p}$

$p$

$p$

$M\otimes \mathbb{R}.$

Theorem A.l. An even lattice with invariants only the following conditions are simultaneously satisfied. 1 $t_{(+)}-t_{(-)}\equiv signqmod 8.$ . 2 $t_{(+)}\geq 0,$

$(t_{(+)}, t_{(-)}, q)$ $\cdot$

$\iota f$

$)$

$)$

$t_{(-)}\geq 0,$ $t_{(+)}+t_{(-)}\geq l(\mathfrak{A}_{q})$

exists

if and

120

APPENDIX

3 which $)$

$(-1)^{t_{(-)}}|\mathfrak{U}_{q}|\equiv\det(K(q_{p}))mod (\mathbb{Z}_{p}^{*})^{2}$

for all odd primes for $p$

is the unique -adic lattice with . the discriminant quadratic$fomq_{p}$ and the and 4 (here $K(q_{2})$ is the unique 2-adic lattice with the discriminant . quadratic $fomq_{2}$ and the $t_{(+)}+t_{(-)}=l(\mathfrak{A}_{q_{p}})$

(here

$K(q_{p})$

$p$

$mnkl(\mathfrak{A}_{q_{p}}))$

$)$

$ q_{2}\neq$

$|\mathfrak{A}_{q}|\equiv\pm\det(K(q_{2}))mod (\mathbb{Z}_{2}^{*})_{l}^{2}ft_{(+)}+t_{(-)}=l(\mathfrak{A}_{q_{2}})$

$q_{\theta}^{(2)}(2)\oplus q_{2}’$

$mnkl(\mathfrak{U}_{q_{2}}))$

From

$l(\mathfrak{A}_{q})=\max_{p}l(\mathfrak{A}_{q_{p}})$

,

we obtain the important corollary.

Corollary A.2. An even lattice with invariants $(t_{(+)},t_{(-)}, q)$ exists following conditions are simultaneously satisfied. 1 $t_{(+)}-t_{(-)}\equiv signqmod 8.$

$\iota f$

the

$\cdot$

$)$

2

$)$

$t_{(+)}\geq 0,$ $t_{(-)}\geq 0,$ $t_{(+)}+t_{(-)}>l(\mathfrak{U}_{q})$

Theorem 1.16.5 and Corollary 1.16.6 in for odd lattices.

. $[Nik80b]$

give similar results

A.1.5. Primitive embeddings into even unimodular lattices We have a simple statement (see Proposition 1.4.1 in

$[Nik80b]$ ).

Proposition A.3. Let $M$ be an even lattice. Its overlattice $M\subset N$ offinite index is equivalent to the isotropic subgroup $H=N/M\subset \mathfrak{U}_{M}$ with respect to . Moreover, we have $q_{N}=q_{M}|(H^{\perp})/H.$ $q_{M}$

An embedding of lattices $M\subset L$ is primitive if $L/M$ is a free module. Let be an even unimodular lattice, $M\subset L$ its primitive sublattice, and $T=M_{L}^{\perp}$ . Then $M\oplus T\subset L$ is an overlattice of a finite index. Applying Proposition A.3, we obtain that $H=L/(M\oplus T)$ is the graph of an isomolphism : $q_{M}\cong-q_{T}$ , and this is equivalent to a primitive embedding $M\subset L$ into an even unimodular lattice with $T=M_{L}^{\perp}$ . Thus we have (see Proposition 1.6.1 in $[Nik80b])$ $\mathbb{Z}-$

$L$

$\gamma$

Proposition A.4. primitive embedding of an even lattice $M$ into an even unimodular lattice, in which the orthogonal complement is $isomo\varphi hic$ to $T$, is detemined by an isomorphism : $q_{M}\cong-q_{T}.$ Two such isomorphisms and determine isomorphic primitive emare conjugate via an automorphism of beddings and only $A$

$\gamma$

$\gamma’$

$\gamma$

$\iota f$

$\iota fth\varphi$

$T.$

From Theorem A. 1 and Corollary A.2 we then obtain (Theorem 1.12.2 and Corollary 1.12.3 in $[Nik80b])$ Theorem A.5. The following pmperties are equivalent: a There exists a primitive embedding of an even lattice $M$ with invari. ants $(t_{(+)}, t_{(-)}, q)$ into some even unimodular lattice ofsignature $)$

$(l_{(+)}, l_{(-)})$

121

A. 1. DISCRIMINANT FORMS TECHNIQUE

b There exists an even lattice with invariants $(l_{(+)}-t_{(+)}, l_{(-)}-t_{(-)}, -q)$ c There exists an even lattice with invariants $(l_{(-)}-t_{(-)}, l_{(+)}-t_{(+)}, q)$ . d The following conditions are simultaneously satisfied $)$

$)$

$)$

1 2 3

$)$

$l_{(+)}-l_{(-)}\equiv 0mod 8.$

$)$

$l_{(+)}-t_{(+)}\geq 0,$ $l_{(-)}-t_{(-)}\geq 0,$ $l_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}\geq l(\mathfrak{U}_{q})$

$)$

$(-1)^{l_{(+)}-t_{(+)}}|\mathfrak{A}_{q}|\equiv\det(K(q_{p}))mod (\mathbb{Z}_{p}^{*})^{2}$

.

for all odd primes for $p$

which $l_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}=l(\mathfrak{A}_{q_{p}})$ (here $K(q_{p})$ is the unique -adic lattice with the discriminant quadratic form and the . 4 (here $K(q_{2})$ is the unique 2-adic lattice with the and . discriminant quadratic form and the $p$

$mnkl(\mathfrak{A}_{q_{p}}))$

$q_{p}$

$)$

$|\mathfrak{A}_{q}|\equiv\pm\det(K(q_{2}))mod (\mathbb{Z}_{2}^{*})^{2}\iota fl_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}=l(\mathfrak{A}_{q_{2}})$

$q_{2}\neq q_{\theta}^{(2)}(2)\oplus q_{2}’$

$q_{2}$

$mnkl(\mathfrak{U}_{q2}))$

Corollary A.6. There exists a primitive embedding of an even lattice $M$ with invariants $(t_{(+)}, t_{(-)}, q)$ into some even unimodular lattice ofsignature the following conditions are simultaneously satisfied. $\cdot$

$(l_{(+)}, l_{(-)})\iota f$

1 2

$)$

$l_{(+)}-l_{(-)}\equiv 0mod 8.$

. It is well-known that an even unimodular lattice of signature . see [Ser70]). The same is valid is unique if it is indefinite ( if $l_{(+)}+l_{(-)}\leq 8$ . Thus, Theorem A.5 and Corollary A.6 give existence of a primitive embedding of $M$ into these unimodular lattices. $)$

$l_{(-)}-t_{(-)}\geq 0,$ $l_{(+)}-t_{(+)}\geq 0,$ $l_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}>l(\mathfrak{U}_{q})$

$(l_{(+)}, l_{(-)})$

$e.$

$g$

A.1.6. Uniqueness

We restrict ourselves to the following uniqueness result (see Theorem 1.14.2 in $[Nik80b]$ and Theorem 1.2’ in $[Nik80a])$ . We note that this is based on fundamental results about spinor genus of indefinite lattices of the rank due to M. Eichler and M. Kneser. $\geq 3$

Theorem A.7. Let $T$ be an even indefinite lattice with the invariants satisfying the following conditions: rk for all $p\neq 2.$ , then or b If rk $T$ Then the lattice is unique (up to isomorphisms), and the canonical homomorphism $O(T)\rightarrow O(q_{T})$ is surjective. $(t_{(+)},$

$t_{(-)},$ $q)$

$a)$

$)$

$T\geq l(\mathfrak{U}_{q_{p}})+2$

$T=l(\mathfrak{U}_{q_{2}})$

$q_{2}\cong u_{+}^{(2)}(2)\oplus q’$

$q_{2}\cong v_{+}^{(2)}(2)\oplus q’.$

Applying additionally Proposition A.4, we obtain the following Analogue of Witt’s Theorem for primitive embeddings into even unimodular lattices (see Theorem 1.14.4. in $[Nik80b]$ ): , and let Theorem A.8. Let $M$ be an even lattice ofsignature . Then a primitive embe an even unimodular lattice ofsignature bedding of $M$ into is unique (up to isomorphisms), pmvided thefollowing conditions hold: $(t_{(+)}, t_{(-)})$

$(l_{(+)}, l_{(-)})$

$L$

$L$

122

APPENDIX

1 2 3

and $l_{(-)}-t_{(-)}>0.$

$)$

$l_{(+)}-t_{(+)}>0$

$)$

$l_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}\geq 2+l(\mathfrak{A}_{M_{p}})$

$)$

If

for all

$l_{(+)}+l_{(-)}-t_{(+)}-t_{(-)}=l(\mathfrak{U}_{M_{2}})$

,

then

$p\neq 2.$ $q_{M}\cong u_{+}^{(2)}(2)\oplus q’$

or

$q_{M}\cong v_{+}^{(2)}(2)\oplus q’.$

A.2. Classification of main invariants and their geometric interpretation Here we apply results of Section A.1 to classify main invariants and of non-symplectic involutions of K3 surfaces and equivalent right DPN surfaces; moreover, we give their geometric intelpretation (types: el(see Section 2.3). We liptic, parabolic, hyperbolic, and invariants also give proofs of results of Section 2.3 which were only cited there. All these results had been obtained in [ $Nik80a$ , Nik $80b$ , Nik79, Nik83, Nik87] and are well-known. We follow these papers. We follow notations and considerations of Section 2.3. According to Section 2.3, the set of main invariants of K3 surfaces with non-symplectic involution is exactly the set of isomorphism classes of $2-elementa\prime y$ even hyperbolic lattices having a primitive embedding $S\subset L$ where $L\cong L_{K3}$ is an even unimodular lattice of signature (3, 19). Further we denote $L=L_{K3}$ . We remind that a lattice $M$ is called 2is 2elementary if its discriminant group elementary. is also 2-elementary, let us more generally Since the lattice the consider all even 2-elementary lattices $M$ . We denote by $M$ $M$ is 2-elementaly, numbers of positive and negative squares of . Since is a 2-elementary group where is the discriminant group its order. We get the important invariant of the discriminant group (and $M$ itself). We have and $a\geq 0.$ of $M$ is a non-degenerate finite quadratic The discriminant form (we call such a form 2-elementaly.) form on the 2-elementaly group is orthogonal By Jordan decomposition (see Section A.1.3), the form with and sum of elementary finite quadratic forms is the signature $0mod 8,4mod 8$ and lmod8 respectively. If , then and is even: it takes sum of only elementary forms . Otherwise it is odd: at least one of its values belongs values only in to $\{-1/2,1/2\}mod 2$ . Therefore, we introduce an important invariant $\delta\in\{0,1\}$ of (and of $M$ itself). The $\delta=0$ if is even, and $\delta=1$ if is odd. $S$

$(r, a, \delta)$

$(k, g, \delta))$

$S$

$S$

$\mathfrak{A}_{M}=M^{*}/M\cong(\mathbb{Z}/2\mathbb{Z})^{a}$

$T=S_{L}^{\perp}$

$(t_{(+)}, t_{(-)})$

$2^{a}$

$\mathfrak{A}_{M}\cong(\mathbb{Z}/2\mathbb{Z})^{a}$

$a$

$\mathfrak{U}_{M}$

$a\in \mathbb{Z}$

$q_{M}$

$\mathfrak{A}_{M}$

$q_{M}$

$u_{+}^{(2)}(2),$

$v_{+}^{(2)}(2)$

$\pm$

$u_{+}^{(2)}(2)$

$q_{M}$

$v_{+}^{(2)}(2)$

$q_{M}$

$\mathbb{Z}/2\mathbb{Z}$

$q_{M}$

$q_{M}$

$q_{\pm 1}^{(2)}(2)$

$q_{M}$

123

A.2. MAIN INVARIANTS

We have the important relations between elementary forms: $2u_{+}^{(2)}(2)\cong 2v_{+}^{(2)}(2),$

$3q_{\pm 1}^{(2)}\cong q_{\mp 1}^{(2)}(2)\oplus v_{+}^{(2)}(2),$

. It follows that

$u_{+}^{(2)}(2)\oplus q_{\pm 1}^{(2)}(2)$

$q_{M}$

can be written in the canonical form de-

and $\sigma pending on its invariants $mod 8$ . We have several cases: $\delta$

$a,$

$\delta=0$

: then

$ a\equiv$

Omod2,

$ q_{1}^{(2)}(2)\oplus q_{-1}^{(2)}(2)\oplus q_{\pm 1}^{(2)}(2)\cong$

mod 8=signq_{M}mod 8\equiv t_{(+)}-t_{(-)}$

$\sigma\equiv 0mod 4$

, and

$\sigma\equiv 0mod 8$

if $a=0$ . We

have $q_{M}\cong sv_{+}^{(2)}(2)\oplus(a/2-s)u_{+}^{(2)}(2)$

where $s=0$ or 1 and $\sigma\equiv 4smod 8.$

: then $a\geq 1,$ $\sigma\equiv amod 2,$ 8$ if $a=2$ . We have

$\delta=1$

$mod

$\sigma\equiv\pm 1mod 8$

if $a=1$ , and

$q_{M}\cong q_{\pm 1}^{(2)}(2)\oplus((a-1)/2)u_{+}^{(2)}(2)if\sigma\equiv\pm 1$

$q_{M}\cong 2q_{\pm 1}^{(2)}\oplus(a/2-1)u_{+}^{(2)}(2)if\sigma\equiv\pm 2$

mod8; mod8;

$q_{M}\cong q_{1}^{(2)}(2)\oplus q_{-1}^{(2)}(2)\oplus(a/2-1)u_{+}^{(2)}(2)if\sigma\equiv 0$

mod8;

$q_{M}\cong q_{1}^{(2)}(2)\oplus q_{-1}^{(2)}(2)\oplus v_{+}^{(2)}(2)\oplus(a/2-2)u_{+}^{(2)}(2)if\sigma\equiv 4$

$(\sigma\equiv t_{(+)}-$

$q_{M}$

$\delta)$

$a,$

$q_{M}$

mod8.

is determined by its invariants we have listed above all conditions of exisfor the given invariants $\sigma=t_{(+)}-t_{(-)}mod 8,$ $a\geq 0$ and

Thus, the discriminant form $t_{(-)}mod 8,$ . Moreover,

tence of

$\sigma\not\equiv 4$

$\delta\in\{0,1\}.$

Assume that these conditions are satisfied. By Corollary A.2, a 2elementary lattice $M$ with invariants $(t_{(+)}, t_{(-)}, q_{M})$ exists if $t_{(+)}\geq 0,$ $t_{(-)}\geq 0$ and $t_{(+)}+t_{(-)}>a=l(\mathfrak{U}_{M})$ . The condition $t_{(+)}+t_{(-)}\geq a$ is necessary for the existence. Assume that $t_{(+)}+t_{(-)}=a$ . If $\delta=1$ , then , and the lattice $M$ also does exist by Theorem A. 1. If $\delta=0$ , then $M(1/2)$ will be an even unimodular lattice. It follows that the condition $t_{(+)}-t_{(-)}\equiv 0mod 8$ must be satisfied, and it is sufficient for the existence of $M$ since an even unimodular lattice $M(1/2)$ with the invariants does exist under this condition. Thus, we finally listed all conditions of existence of an even -elementaly lattice $M$ with the invariants $q_{M}\cong q_{\pm 1}^{(2)}(2)\oplus q’$

$(t_{(+)}, t_{(-)})$

$2$

$(t_{(+)}, t_{(-)}, a, \delta)$

.

we had proved that the invariants

define the if a prime is of . We have discriminant quadratic form . By Theorem A.7, or odd. If $a\geq 3$ , then then the lattice $M$ is unique if it is indefinite and rk $M\geq 3$ . Moreover, then is epimorphic. If rk $ M\leq$ the canonical homomorphism $U$ , then $M$ is one of lattices: or $U(2)$ . One can easily check for them the same statements directly. Moreover,

$q_{M}$

$M$

$q_{M}\cong u_{+}^{(2)}(2)\oplus q’$

$l(\mathfrak{U}_{M_{p}})=0$

$q_{M}\cong v_{+}^{(2)}(2)\oplus q’$

$O(\Lambda l)\rightarrow O(q_{M})$

$2$

$\langle\pm 2\rangle,$

$(t_{(+)}, t_{(-)}, a, \delta)$

$\langle\pm 2\rangle\oplus\langle\pm 2\rangle,$

$p$

124

APPENDIX

we get the following classification result about 2-elementary even indefinite (except few exceptions) lattices. It is Theorems 3.6.2 and 3.6.3 from $[Nik80b].$ Thus, finally

Theorem A.9. The genus of an even 2-elementary lattice $M$ is determined ; and either $M$ is indefinite or rk $M=2,$ by the invariants these invariants determine the isomorphism class of $M$, and the canonical $homomo\varphi hismO(M)\rightarrow O(q_{M})$ is epimorphic. exists An even $ 2-elementa\eta$ lattice $M$ with invariants and only ifall the following conditions are satisfied (it being assumed that $\delta=0$ $t_{(-)}\geq 0$): or 1, and that $(t_{(+)}, t_{(-)}, a, \delta)$

$\iota f$

$(t_{(+)}, t_{(-)}, a, \delta)$

$\iota f$

$a,$ $t_{(+)},$

1 2 3 4 5

;

$)$

$a\leq t_{(+)}+t_{(-)}$

$)$

$t_{(+)}+t_{(-)}+a\equiv 0mod 2$ ;

$)$

$t_{(+)}-t_{(-)}\equiv 0mod 4\iota f\delta=0$

$)$

$(\delta=0, t_{(+)}-t_{(-)}\equiv 0mod 8)_{l}fa=0$ ;

$)$

$t_{(+)}-t_{(-)}\equiv\pm 1mod 8$

;

if $a=1$ ;

$6)\delta=0\iota f(a=2, t_{(+)}-t_{(-)}\equiv 4 mod 8)$ ;

7

$)$

$t_{(+)}-t_{(-)}\equiv 0mod 8\iota f(\delta=0, a=t_{(+)}+t_{(-)})$

.

Let be a main invaniant and $r=$ rk . Since is 2-elementary even hyperbolic, by Theorem A.9 it is then detennined by its invariants $(t_{(+)}=$ $S$

$S$

$1,t_{(-)}=r-1,$

$a,$

$\delta)$

$S$

.

is equivBy Theorem A.5, existence of a primitive embedding $T=S^{\perp}$ with invariants alent to existence of a 2-elementary even lattice $(t_{(+)}=2, t_{(-)}=20-r, a, \delta)$ (indeed, $q_{T}\cong-q_{S}$ has the same invariants and . Thus, the set of main invariants is equal to the set of such r-1, a, \delta)$ $(1, $(2, 20-r, a, \delta)$ that both and satisff conditions ) $-7)$ of which are presented in Figure Theorem A.9. It consists of exactly $S\subset L_{K3}$

$a$

$\delta)$

$S$

$(r, a, \delta)$ $1$

$(r, a, \delta)$

1.

is uniquely deBy Theorem A.9, the orthogonal complement , and the canonical homomorphism $O(T)\rightarrow O(q_{T})$ termined by is epimorphic. By Proposition A.4 the primitive embedding $S\subset L_{K3}$ is unique up to automorphisms of $L_{K3}.$ $T=S_{L}^{\perp}$

$(r, a, \delta)$

Let us show that $O(S\subset L_{K3})$ contains an automorphism of spinor norm $-1(i.$ . it changes two connected components of the quadric , see , or (30) . Using Theorem A.9, it is easy to see that either $T=U\oplus T’$ , or $T=U(2)\oplus T’$ , or . If $T$ $+1$ on the first and-l we consider the automorphism of which is $on$ the second . In remaining cases we consider an automorphism of $T$ which is-l on the first 2-dimensional hyperbolic summand of $T$ , and which is $+1$ on $T’$ . It is easy to see that changes connected components of the . On the other hand, is identical on the $\tau*/T$ and can be $\Omega_{S\subset L_{K3}}$

$e$

$ T=\langle 2\rangle\oplus\langle 2\rangle$

$)$

$T=\langle 2\rangle\oplus\langle-2\rangle\oplus T’$

$\langle 2\rangle$

$\alpha$

$\alpha$

$\langle 2\rangle$

$\alpha$

$\Omega_{S\subset L_{K3}}$

$T\cong\langle 2\rangle\oplus\langle 2\rangle,$

$\alpha$

125

A.2. MAIN INVARIANTS

continued identically on . This extension gives an automorphism of which is identical on and has spinor norm-l. It follows (see Section 2.3) that for the fixed main invariants the moduli space Mod of K3 surfaces with non-symplectic involution is irreducible. Now let us consider the geometric interpretation of main invariants in terms of the set $C=X^{\theta}$ of the fixed points. The set $C$ is non-singular. Indeed, if $x\in C$ a singular point of $C$ , then is the identity in the tangent space . Then for any $\omega_{X}\in H^{2,0}(X)$ and is symplectic. We get a contradiction. For a non-singular irreducible curve $C$ on a K3 surface $X$ we have $g(C)=(C^{2}+C\cdot K_{X})/2+1=C^{2}/2+1\geq 0$ . It follows that $C^{2}>0$ , if $g(C)>1$ . Since the Picard lattice is hyperbolic, it follows that any two $X$ genus curves on of must intersect. It then follows that has one of types A, listed below: Case $A:X^{\theta}=C_{g}+E_{1}+\cdots+E_{k}$ where is a non-singular irreducible , the curves are non-singular curve of genus $g\geq 0$ and are disjoint to each irreducible rational $(i. e. E_{i}^{2}=-2)$ . All curves $S$

$L_{K3}$

$S$

$(r, a, \delta)$

$(r,a,\delta)$

$(r,$ $a,$

$\delta)$

$\theta$

$\theta$

$\theta^{*}(\omega_{X})=\omega_{X}$

$T_{x}$

$S_{X}$

$X^{\theta}$

$\geq 2$

$B,$ $C$

$C_{g}$

$E_{k}$

$E_{1},$

$ C_{g}\neq\emptyset$

$\ldots,$

$C_{g},$

$E_{i}$

other.

Case $B:X^{\theta}=C_{1}^{(1)}+\cdots+C_{1}^{(m)}+E_{1}+\cdots+E_{k}$ where is disjoint union of $m>1$ elliptic curves (we shall prove in a moment that actually $m=2$ and $k=0$ ). Case By Lefschetz formula, the Euler characteristics $\chi(X^{\theta})=2+r-(22-$ $C_{1}^{(1)}+\cdots+C_{1}^{(m)}$

$C:X^{\theta}=\emptyset.$

$r)=2r-20.$

By Smith Theory (see [Kha76]), the total Betti number over isfies $\dim H^{*}(X^{\theta}, \mathbb{Z}/2\mathbb{Z})=\dim H^{*}(X, \mathbb{Z}/2\mathbb{Z})-2a=24-2a$ if

$\mathbb{Z}/2\mathbb{Z}$

sat-

$X^{\theta}\neq\emptyset.$

For any 2-dimensional cycle $Z\subset X$ one evidently has $mod 2$ . Thus, $X^{\theta}\sim 0mod 2$ in if and only if $mod 2$ for any $x\in H^{2}(X, \mathbb{Z})$ . Let us write $x\in L=H^{2}(X, \mathbb{Z})$ as $x=$ . Then $x++x_{-}$ where $x+\in S^{*}$ and Moreover, $x^{2}=x_{+}^{2}+x^{\underline{2}}\equiv 0mod 2$ because is even. Taking the $\theta^{*}(x)\equiv 2x_{+}^{2}mod 2$ . sum, we get is unimodular, Since any $x+\in S^{*}$ appears in this identity. It follows that $X^{\theta}\sim 0mod 2$ for any $x+\in S^{*}$ . Equivalently, the if and only if in invariant $\delta=0$ . Therefore $Z\cdot\theta(Z)\equiv Z\cdot X^{\theta}$

$(x, \theta^{*}x)\equiv 0$

$H^{2}(X, \mathbb{Z})$

$x_{-}\in(S^{\perp})^{*}$

$x\cdot\theta^{*}(x)=x_{+}^{2}-x^{\underline{2}}.$

$L$

$H^{2}(X, \mathbb{Z})$

$x$

$H^{2}(X, \mathbb{Z})$

$\delta=0$

$x_{+}^{2}\in \mathbb{Z}$

if and only if

$X^{\theta}\sim 0$

$mod 2$

in

$H^{2}(X, \mathbb{Z})$

.

belong to one elliptic pencil $|C|$ of elliptic In case , elliptic curves curves where it is known (see [PS-Sh71]) that $C$ is primitive in Picard lattice $B$

$C_{1}^{(i)}$

126

APPENDIX

.

Assume that either $m>2$ or $k>0$ . Then is trivial on the base of the elliptic pencil. Since it is also trivial on a fibre which is not multiple, is symplectic, and we get a contradiction. Thus, in Case we $S_{X}$

$\mathbb{P}^{1}$

$\theta$

$C_{1}^{(i)}$

$B$

$\theta$

have $k=0,$ $m=2$ and $\delta=0.$ In case , the quotient $Y=X/\{1, \theta\}$ is an Enriques surface. It follows that $r=a=10$ and $\delta=0$ in this case. Combining all these arguments, we obtain the geometric interpretation of the invariants cited in Section 2.3 $C$

$(r, a,\delta)$

A.3. The analogue of Witt’s theorem for 2-elementary finite forms Here we follow Section 1.9 in $[Nik84b]$ to prove an important Lemma 2.7. We consider a 2-elementary finite bilinear forms : and 2-elementaly finite quadratic forms : on finite 2-elementaly $b$

$q$

groups $B,$

$B\times B\rightarrow\frac{1}{2}\mathbb{Z}/\mathbb{Z}$

$Q\rightarrow\frac{1}{2}\mathbb{Z}/2\mathbb{Z}$

$Q.$

In the previous section we gave classification of non-degenerate 2-elementary finite quadratic forms. Similarly one can classify non-degenerate 2-elementaly finite bilinear forms. They are orthogonal sums of elementary . The form fonns and is the bilinear form of quadratic , and the form fonns is the bilinear fonn of quadratic fonns . We denote by the characteristic element of , i. e. and $b(x, x)=b(s_{b}, x)$ for all $x\in B$ . It is easy to see that any non-degenerate 2-elementaly finite bilinear form is $b_{1}^{(2)}(2)$

$u_{-}^{(2)}(2)$

$q_{\pm}^{(2)}(2)$

$u_{+}^{(2)}(2)$

$b_{1}^{(2)}(2)$

$u_{-}^{(2)}(2)$

$v_{+}^{(2)}(2)$

$b$

$s_{b}$

$b$

$b\cong mu_{-}^{(2)}(2)$

if $b(x, x)=0$ for all $x\in B$ (equivalently the characteristic element $s_{b}=0,$ these bilinear forms are the same as skew-symmetric ones); $b\cong b_{1}^{(2)}(2)\oplus mu_{-}^{(2)}(2)$

,

if $b(s_{b}, s_{b})=\frac{1}{2}mod 1$ ; $b\cong 2b_{1}^{(2)}\oplus mu_{-}^{(2)}(2)$

if

but $b(s_{b}, s_{b})=0.$ We prove (see Section 1.9 in

$s_{b}\neq 0$

$[Nik84b]$ )

Proposition A.10. Let be a non-degenemte bilinear form on a finite 2be an isomorphism of subgmups of elementary gmup $ Band\theta$ : $B$ which preserves the and restrictions and that maps the chamcteristic element of the form to itself $(if ofcourse, it belongs to H_{1})$ . Then extends to an automorphism of $b$

$H_{1}\rightarrow H_{2}$

$b|H_{1}$

$b$

$\theta$

$b.$

$b|H_{2}$

127

A.3. ANALOGUE OF $WIT\Gamma’ S$ THEOREM

Proposition A.ll. Let be a quadraticform on afinite $ $Q$ whose kemel is $zem$ ; that is $q$

$\{x\in Q|x\perp Q$

2-elementa\eta$

gmup

and $q(x)=0\}=0.$

Let : be an isomorphism of two subgroups of $Q$ that preserves the restrictions and and that maps the elements of the kemel and the chamcteristic elements of the bilinear form into the same sort of elements $(ofcourse, fthey belong to H_{J})$ . Then extends to an automorphism $\theta$

$H_{1}\rightarrow H_{2}$

$q|H_{2}$

$q|H_{1}$

$q$

$\phi$

of

$q$

$(an element s\in Q is called characteristic \iota fq(s, x)=q(x, x)$

$x\in Q)$

for all

.

We shall prove the propositions by induction on the number of generators of $B$ and $Q$ . Let us begin with Proposition A.10. Suppose there exist and $x_{2}=\theta(x_{1})\in H_{2}$ such that $b(x_{1}, x_{1})=b(x_{2}, x_{2})=\frac{1}{2}mod 1.$ $b_{1}=b|B_{1},$ $b_{2}=b|B_{2},$ Write . Then the same conditions hold for the and , their and nondegenerate forms and defined on the subgroups : , and an isomorphism and subgroups and are characteristic siEverything reduces to extending . Since (this follows from classification of multaneously, and are isomorphic Therefore, forms). the existence of an extension of nondegenerate bilinear follows from the induction hypothesis. To complete the proof it remains is zero. Deand to consider the case when the function $b(x, x)$ on $B$ . It is easy to check (using the note by the characteristic element of classification again) that the natural homomorphism $x_{1}\in H_{1}$

$B_{2}=(x_{2})_{B}^{\perp},$

$B_{1}=(x_{1})_{B}^{\perp},$

$H_{1}’=(x_{1})_{H_{1}}^{\perp},$

$\theta’=\theta|H_{1}$

$H_{2}’=(x_{2})_{H_{2}}^{\perp}$

$B_{1}$

$b_{2}$

$b_{1}$

$\theta’$

$H_{2}’\subset B_{2}$

$H_{1}’\subset B_{1}$

$B_{2}$

$H_{1}’\cong H_{2}’.$

$\theta’$

$x_{2}$

$x_{1}$

$b_{1}$

$b_{2}$

$\theta’$

$H_{1}$

$H_{2}$

$s$

$O(b)\rightarrow O(s^{\perp})$

is epimorphic (we always consider a subgroup with the restriction of the ; therefore, it lie in and form on the subgroup). In our case , this is obvious suffices to extend to an automolphism of . If since in this case is a nondegenerate skew-symmetric fonn; for them the , then is the kemel of proposition is well-known and obvious. If , and, by the hypothesis, $b$

$s^{\perp}$

$\theta$

$s^{\perp}$

$H_{2}$

$H_{1}$

$s\not\in s^{\perp}$

$s^{\perp}$

$s\in s^{\perp}$

$[s]$

$s^{\perp}$

$[s]\cap H_{1}=\theta([s]\cap H_{1})=[s]\cap H_{2}.$

Let $\overline{\theta}:H_{1}/([s]\cap H_{1})\cong H_{2}/([s]\cap H_{2})$

is nondegenerbe the isomorphism $\theta mod [s]\cap H_{1}$ . Then, because . Let ate and skew-symmetric, extends to an automorphism $\psi(x)-\theta(x)=\overline{g}(x)s,$ . Then be a lifting to an automorphism of $x\in H_{1},$ is a linear function. Extending to a if where : $s^{\perp}/[s]$

$\overline{\psi}\in O(s^{\perp}/[s])$

$\overline{\theta}$

$\psi$

$(s)^{\perp}$

$of\overline{\psi}$

$\overline{g}$

$H_{1}\rightarrow \mathbb{Z}/2\mathbb{Z}$

$\overline{g}$

128

APPENDIX

linear function

$g$

:

$s^{\perp}\rightarrow \mathbb{Z}/2\mathbb{Z}$

,

we put

$\tilde{\psi}(x)=\psi(x)+\tilde{f}(x)sifx\in s^{\perp}$

is the desired extension of Evidently Let us prove Proposition A. 11. Assume that the bilinear form of has a nonzero kemel. Then it is generated by an element , and $q(r)=1$ $mod 2$ . Using Proposition A.10, we can extend to an automorphism of $f(x)=q(x)-q(\psi(x))\in \mathbb{Z}/2\mathbb{Z}$ , where the bilinear fonn of . The function $x\in Q$ , is linear and vanishes on . Evidently, $\tilde{\psi}(x)=\psi(x)+f(x)r,$ where $x\in Q$ , is the desired extension of It remains to examine the case when the bilinear fonn is nondegenerate. The case where there exist elements $x_{1}\in H_{1}$ and $x_{2}=\theta(x_{1})\in H_{2}$ , can be examined similarly for which $ q(x_{1})=q(x_{2})=\frac{1}{2}\theta$ where to the corresponding case of Proposition A.10. Therefore, we assume that Omodl if $x\in H_{1}$ . For a characteristic element $s\in Q$ the natural homomorphism $\tilde{\psi}\in O(s^{\perp})$

$\theta.$

$q$

$r$

$\theta$

$\psi$

$q$

$H_{1}$

$\theta.$

$q$

$\theta\not\in 2\mathbb{Z}$

$ q(x)\equiv$

$O(q)\rightarrow O(s^{\perp})$

is epimorphic (this easily follows from the classification of nondegenerate quadratic forms on 2-elementaly groups given in Section A.2). In our case lie in , and it suffices to extend to an automorphism of and is non-degenerate, this follows from the classical If the bilinear form on Witt theorem over the field with two elements. Suppose it is degenerate; is generated by . In the case $q(s)=0$ , one can then the kemel of pass to a form on and we argue in the same way as in the proof of Proposition A.10. But if $q(s)\equiv 1mod 2$ , then we pass to the first case, already treated. Proposition A. 11 is proved. $H_{1}$

$H_{2}$

$s^{\perp}$

$s^{\perp}.$

$\theta$

$s^{\perp}$

$s^{\perp}$

$s$

$s^{\perp}/[s]$

A.4. Calculations of fundamental chambers Here we outline calculations of fundamental chambers for hyperbolic reflection groups which had been used in the main part of the work. of 2-elementary even A.4.1. Fundamental chambers hyperbolic lattices of elliptic type (Table 1). $\lambda 4^{(2,4)}$

We consider a1150 types of 2-elementary even hyperbolic lattices of el. We outline the calculation liptic type given by their full invariants (equivalently, the corresponding Dynkin of a fundamental chamber for the full reflection group $W^{(2,4)}(S)=W(S)$ . diagram in all roots of . They are This is the group generated by reflections $S$

$(r, a, \delta)$

$\mathcal{M}^{(2,4)}$

$\Gamma(P(\mathcal{M}^{(2,4)})))$

$s_{f}$

$f$

$S$

129

A.4. CALCULATIONS OF FUNDAMENTAL CHAMBERS

elements $f\in S$ either with $f^{2}=-2$ or with $f^{2}=-4$ and $mod 2$ . The reflection $s_{f}\in O(S)$ is then given by

$(f, S)\equiv 0$

$x\mapsto x-\frac{2(x,f)f}{f^{2}}, \forall x\in S.$

We use Vinberg’s algorithm [Vin72] which we describe below. It can be applied to any hyperbolic lattice and any of its reflection subgroup $W\subset W(S)$ which is generated by reflections in some precisely described subset of primitive roots of which is $W-inva\dot{n}ant.$ First, we should choose a non-zero $H\in S$ with $H^{2}\geq 0$ . Then $H$ defines the half cone $V^{+}(S)$ such that $H\in V^{+}(S)$ . We want to find a fundamental chamber of $W$ containing of all roots from which are orthogStep . We consider the subset $H$ onal to . This set is either a finite root system or affine root system. One should choose a bases in . For example, one can take another element $H_{1}\in S$ such that $(H, H_{1})>0$ and does not contain zero. Then $S$

$S$

$\triangle\subset S$

$\mathbb{R}^{+}h.$

$\mathcal{M}\subset \mathcal{L}(S)=V^{+}(S)/\mathbb{R}^{+}$

$0$

$\triangle$

$\Delta_{0}$

$P_{0}$

$\triangle_{0}$

$H_{1}^{2}\geq 0,$

$(H_{1}, \triangle_{0})$

$\triangle_{0}^{+}=\{f\in\triangle_{0}|(f, H_{1})>0\},$

and

$P_{0}\subset\triangle_{0}^{+}$

others. For

which are not non-trivial sums of consists of roots from with $(f, H)\geq 0$ we introduce the height $\triangle_{0}^{+}$

$ f\in\triangle$

$h(f)=\frac{2(f,H)^{2}}{-f^{2}}.$

The height is equivalent to the hyperbolic distance between the point and the hyperplane which is orthogonal to . The set of all possible heights is a discrete ordered subset

$\mathbb{R}^{+}H$

$f$

$\mathcal{H}_{f}$

(90)

$h_{0}=0, h_{1}, h_{2}, \ldots h_{i}, \ldots,$

. It is always a subset of non-negative integers, and one can always of as the set of possible heights. take The fundamental chamber is defined by the set of orthogonal roots to which is $\mathbb{R}^{+}$

$\mathbb{Z}^{+}$

$\mathcal{M}\subset \mathcal{L}(S)$

$ P(\mathcal{M})\subset\Delta$

$\mathcal{M}$

(91)

$P(\mathcal{M})=\bigcup_{0\leq j}P_{j}$

is defined above and for $j>0$ consists of all H)>0$ , the height $h(f)=h_{j}$ , and

where $(f,

(92)

$P_{0}$

$P_{j}$

$(f,\bigcup_{0\leq i\leq j-1}P_{i})\geq 0.$

Then (93)

$\mathcal{M}=\{\mathbb{R}^{+}x\in \mathcal{L}(S)|(x, P(\mathcal{M}))\geq 0\}.$

$ f\in\Delta$

such that

130

APPENDIX

If has finite volume, the algorithm terminates after a finite number $m$ of steps . all are empty for $j>m$ , and $\mathcal{M}$

$(i.$

$P_{j}$

$e$

(94)

$P(\mathcal{M})=\bigcup_{0\leq j\leq m}P_{j}$

whenever (93) defines a polyhedron given by (94).

$\mathcal{M}$

of finite volume in

$\mathcal{L}(S)$

for

$P(\mathcal{M})$

Below we apply this algorithm to 2-elementaly hyperbolic lattices of consists of all elliptic type and $W=W(S)=W^{(2,4)}(S)$ . The set $f\in S$ such that either $f^{2}=-2$ or $f^{2}=-4$ and $(f, S)\equiv$ Omod2. $S$

$\Delta\subset S$

Cases

$S=\langle 2\rangle\oplus lA_{1}$

$0\leq l\leq 8$

.

where $0\leq l\leq 8$ Then

$(r, a, \delta)=(1+l, 1+l, 1)$ ,

. We use the standard orthogonal basis

and the standard orthogonal basis

$v_{1},$

$\ldots,$

$v_{l}$

for

for where $h^{2}=2,$ where $v_{1}^{2}=\cdots=v_{l}^{2}=$

$h$

$lA_{1}$

$\langle 2\rangle$

$-2.$

We take $H=h,$ $H_{1}=th+v_{1}+2v_{2}+\cdots+lv_{l}$ where $t>>0$ . Then consists of roots: $\beta_{0}=h-v_{1}-v_{2}$ if $l=2;\sqrt{}0=h-v_{1}-v_{2}-v_{3}$ if if $l\geq 1.$ if $l\geq 3;\sqrt{}1=v_{1}-v_{2},$ is obviously a simplex in For $2\leq l\leq 8$ the polyhedron gives the Dynkin diagrams of of finite volume. The Gram matrix cases $N=1,$ $N=3-10$ of Table 1. Here we repeated calculations by Vinberg in [Vin72]. $P(\mathcal{M}^{(2,4)})$

$l\geq 2;\sqrt{}\iota=v_{l}$

$\sqrt{}\iota_{-1}=v_{l-1}-v_{l}$

$\ldots,$

$\mathcal{M}^{(2,4)}$

$\mathcal{L}(S)$

$(\beta_{i}, \beta_{j})$

. Then

if $l=0$ , and $(r, a, \delta)=(2+l, l, 1)$ if $1\leq l\leq 8$ . We use the standard basis for $U$ where $c_{1}^{2}=c_{2}^{2}=0$ and $(c_{1}, c_{2})=1$ , and the standard olthogonal basis as above. for if $l\geq 1$ ; We take $H=c_{1}$ . We can take which consists of $\beta_{l-1}=v_{l-1}-v_{l}$ if if $l\geq 1$ . Then $e=-c_{1}+c_{2}$ and additional elements $\gamma_{1}=2c_{1}+2c_{2}-v_{1}-$ consists of $v_{2}-v_{3}-v_{4}-v_{5}$ if $l=5$ , and $\gamma_{1}=2c_{1}+2c_{2}-v_{1}-v_{2}-v_{3}-v_{4}-v_{5}-v_{6}$ if $l\geq 6.$ later. We obtain the We shall discuss the finiteness of volume of $N=15$ of Table 1. diagrams of cases $N=11-20$ excluding

Cases

$S=U\oplus lA_{1},0\leq l\leq 8$

$(r, a, \delta)=(2,0,0)$

$c_{1},$

$v_{1},$

$\ldots,$

$c_{2}$

$lA_{1}$

$v_{l}$

$\sqrt{}0=c_{1}-v_{1}$

$P_{0}$

$P(\mathcal{M}^{(2,4)})$

$l\geq 2;\sqrt{}\iota=v_{l}$

$\beta_{1}=v_{1}-v_{2},$

$\ldots,$

$P_{0},$

$\mathcal{M}^{(2,4)}$

Cases

. Then

if $l=0,$ and $(r, a, \delta)=(6+l, 2+l, 1)$ if $1\leq l\leq 5$ . We use the standard bases for as above. We use the standard orthogonal basis for $U$ and ; the lattice where for consists and $x_{1}+\cdots+x_{m}\equiv 0mod 2.$ where of all We take $H=c_{1}$ , and we can take which consists of $S=U\oplus D_{4}\oplus lA_{1},0\leq l\leq 5$

$(r, a, \delta)=(6,2,0)$

$c_{1},$

$v_{1},$

$\epsilon_{m}$

$\epsilon_{1},$

$v_{l}$

$c_{2}$

$lA_{1}$

$\ldots,$

$\epsilon_{1}^{2}=\cdots=\epsilon_{m}^{2}=-1$

$D_{m}\otimes \mathbb{Q}$

$D_{m}$

$\ldots,$

$x_{1}\epsilon_{1}+\cdots+x_{m}\epsilon_{m}$

$x_{i}\in \mathbb{Z}$

$P_{0}$

$\alpha_{1}=\epsilon_{1}-\epsilon_{2},$

$\alpha_{2}=\epsilon_{2}-\epsilon_{3},$

if $l\geq 1;\sqrt{}1=v_{1}-v_{2},$

$\sqrt{}\iota_{-1}=v_{l-1}-v_{l}$ $\ldots,$

$\alpha_{0}=c_{1}-\epsilon_{1}-\epsilon_{2},$

$\alpha_{3}=-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4},$

$\alpha_{4}=2\epsilon_{4},$

$\beta_{0}=c_{1}-v_{1}$

if $l\geq 2;\beta_{l}=v_{t}$ if $l\geq 1.$

131

A.4. CALCULATIONS OF FUNDAMENTAL CHAMBERS

Then

$P(\mathcal{M}^{(2,4)})$

$e=-c_{1}+c_{2},$ consists of $l=3$ , and if $P_{0},$

$\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-$

$\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4}-$

$\epsilon_{2}-\epsilon_{3}-\epsilon_{4}-v_{1}-v_{2}-v_{3}$

if $l\geq 4;\gamma_{2}=2c_{1}+2c_{2}-v_{1}-v_{2}-v_{3}-v_{4}-v_{5}$ if $l=5.$ We obtain the diagrams of cases $N=21-27$ excluding $N=25$ of Table 1. $v_{1}-v_{2}-v_{3}-v_{4}$

Cases

where $m\equiv 0mod 2,$ $m\geq 6,$ $l\geq 0,$ $m+2l\leq 14$ Then $r=2+m+l,$ $a=l+2$ . Moreover $\delta=0$ if $l=0$ and Omod4, otherwise $\delta=1.$ We use the standard bases , and for $U$ , and for for as above. We take $H=c_{1}$ , and we can take which consists of $\alpha_{m}=2\epsilon_{m};\beta_{0}=c_{1}-v_{1}$ if $l\geq 1$ ; $S=U\oplus D_{m}\oplus lA_{1}$

.

$ m\equiv$

$c_{1},$

$v_{1},$

$\ldots,$

$c_{2}$

$D_{m}\otimes \mathbb{Q}$

$\epsilon_{m}$

$\epsilon_{1},$

$\ldots,$

$lA_{1}$

$v_{l}$

$P_{0}$

$\alpha_{1}=\epsilon_{1}-\epsilon_{2},$

$\ldots,$

$\alpha_{0}=c_{1}-\epsilon_{1}-\epsilon_{2},$

$\alpha_{m-1}=\epsilon_{m-1}-\epsilon_{m},$

if

if $l\geq 1.$ Then $e=-c_{1}+c_{2}$ , and some additional consists of elements depending on $m\geq 6$ and $l\geq 0$ where we always assume that $m+2l\leq 14$ and Omod2. $\beta_{1}=v_{1}-v_{2},$

$\ldots,$

$\beta_{l-1}=v_{l-1}-v_{l}$

$P(\mathcal{M}^{(2,4)})$

$l\geq 2;\beta_{l}=v_{l}$

$P_{0},$

$\gamma_{i}$

$ m\equiv$

If $m=6$ and $l=2$ , one must add If $m=6$ and $l=3$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{6}-v_{1}-$

$\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{6}-v_{1}-v_{2}.$

$v_{2}-v_{3},$ $\gamma_{2}=2c_{1}+2c_{2}-2\epsilon_{1}-v_{1}-v_{2}-v_{3}.$

If $m=6$ and $l=4$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{6}-v_{1}-$ $v_{2}-v_{3},$ $\gamma_{2}=2c_{1}+2c_{2}-2\epsilon_{1}-v_{1}-v_{2}-v_{3}-v_{4}.$

If $m=8$ and $l=1$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{8}-v_{1}.$ If $m=8$ and $l=2$ , one must add If $m=8$ and $l=3$ , one must add

$\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{8}-v_{1}-v_{2}.$

$\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{8}-v_{1}-v_{2},$

$\gamma_{2}=2c_{1}+2c_{2}-\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4}-v_{1}-v_{2}-v_{3},$ $\gamma_{3}=2c_{1}+2c_{2}-2\epsilon_{1}-$

$v_{1}-v_{2}-v_{3},$

If $m=10$ and $l=0$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{10}.$ If $m=10$ and $l=1$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{10}-v_{1}.$ If $m=10$ and $l=2$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{10}-v_{1},$ $\gamma_{2}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{6}-v_{1}-v_{2},$ $\gamma_{3}=4c_{1}+4c_{2}-3\epsilon_{1}-\epsilon_{2}-\cdots-$ $\epsilon_{10}-2v_{1}-2v_{2}.$

If $m=12$ and $l=0$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{12}.$ If $m=12$ and $l=1$ , one must add $\gamma_{1}=2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{12},$

$\gamma_{2}=$

$2c_{1}+2c_{2}-\epsilon_{1}-\cdots-\epsilon_{8}-v_{1},$ $\gamma_{3}=6c_{1}+6c_{2}-4\epsilon_{1}-2\epsilon_{2}-2\epsilon_{3}-\cdots-2\epsilon_{11}-3v_{1}.$

We obtain the diagrams for $N=28-50$ of Table 1 except $N=$ $30,34,40,41,44,49,50$ when either $\delta=0$ and $a>2$ , or $a\leq 1.$ Cases

.

where Omod4and $0\leq m\leq 12$ Then $(r, a, \delta)=(2+m, 4,0)$ . We use the standard bases for $U(2)$ where $c_{1}^{2}=c_{2}^{2}=0$ and $(c_{1}, c_{2})=2$ , and the standard basis for above. as We use $H=c_{1}$ and denote $e=-c_{1}+c_{2}$ with $e^{2}=-4.$ $S=U(2)\oplus D_{m}$

$ m\equiv$

$c_{1},$

$c_{2}$

$\epsilon_{m}$

$\epsilon_{1},$

$\ldots,$

$D_{m}\otimes \mathbb{Q}$

132

APPENDIX

If $m=0$ , then If $m=4$ , then $\alpha_{2}=\epsilon_{2}-\epsilon_{3},$

of

$P_{0}$

and

and consists of

$P_{0}$

$e.$

$\alpha_{1}=\epsilon_{1}-\epsilon_{2},$

$\alpha_{0}=c_{1}-\epsilon_{1}-\epsilon_{2}-\epsilon_{3}-\epsilon_{4},$

$\alpha_{3}=-\epsilon_{1}+\epsilon_{2}+\epsilon_{3}-\epsilon_{4},$

$\alpha_{4}=2\epsilon_{4}$

. Then

$P(\mathcal{M}^{(2,4)})$

consists

$e.$

If $m\geq 8$ , then

$P_{0}$

consists of

$\alpha_{0}=c_{1}-2\epsilon_{1},$

$\alpha_{1}=\epsilon_{1}-\epsilon_{2},$

$\ldots,$

$\alpha_{m-1}=$

$\alpha_{m}=2\epsilon_{m}.$

$\epsilon_{m-1}-\epsilon_{m},$

If $m=8$ , then If $m=12$ , then

$P(\mathcal{M}^{(2,4)})$

...

consists of

$P(\mathcal{M}^{(2,4)})$

$ P_{0}=\emptyset$

consists of consists of

$P_{0},$ $e$

$P(\mathcal{M}^{(2,4)})$

and and

$\gamma_{1}=c_{1}+c_{2}-\epsilon_{1}-\cdots-\epsilon_{8}.$

$P_{0},$

$e$

$\gamma_{1}=2c_{1}+c_{2}-\epsilon_{1}-$

$\gamma_{2}=c_{1}+c_{2}-\epsilon_{1}-\cdots-\epsilon_{6}.$

$-\epsilon_{12},$

We obtain the diagrams for $N=2,15,30,44$ of Table 1.

Case $U(2)\oplus D_{4}\oplus D_{4}$ . Then $(r, a, \delta)=(10,6,0)$ . We use standard , ..., for , ..., for the first , and for $U(2)$ and bases the second We take $H=c_{1}$ and which consists of $\epsilon_{4}^{(1)}$

$\epsilon_{1}^{(1)}$

$c_{1},$

$\epsilon_{4}^{(2)}$

$\epsilon_{1}^{(2)}$

$D_{4}$

$c_{2}$

$D_{4}.$

$\alpha_{0}^{(1)}=c_{1}-\epsilon_{1}^{(1)}-\epsilon_{2}^{(1)}-\epsilon_{3}^{(1)}-$

$P_{0}$

$\epsilon_{4}^{(1)},$

$\alpha_{1}^{(1)}=\epsilon_{1}^{(1)}-\epsilon_{2}^{(1)},$

$\alpha_{4}^{(1)}=2\epsilon_{4}^{(1)}$

and

$\alpha_{2}^{(2)}=\epsilon_{2}^{(2)}-\epsilon_{3}^{(2)},$

$\alpha_{2}^{(1)}=\epsilon_{2}^{(1)}-\epsilon_{3}^{(1)},$

$\alpha_{3}^{(1)}=-\epsilon_{1}^{(1)}+\epsilon_{2}^{(1)}+\epsilon_{3}^{(1)}-\epsilon_{4}^{(1)},$

$\alpha_{0}^{(2)}=c_{1}-\epsilon_{1}^{(2)}-\epsilon_{2}^{(2)}-\epsilon_{3}^{(2)}-\epsilon_{4}^{(2)},$

$\alpha_{3}^{(2)}=-\epsilon_{1}^{(2)}+\epsilon_{2}^{(2)}+\epsilon_{3}^{(2)}-\epsilon_{4}^{(2)},$

$\alpha_{1}^{(2)}=\epsilon_{1}^{(2)}-\epsilon_{2}^{(2)},$

$\alpha_{4}^{(2)}=2\epsilon_{4}^{(2)}.$

consists of and $e=-c_{1}+c_{2}.$ Then We obtain the diagram for $N=25$ of Table 1. $P(\mathcal{M}^{(2,4)})$

$P_{0}$

Cases $S=U\oplus E_{7},$ Respectively $(r, a, \delta)=(9,1,1),$ $(10,0,0),$ $(11,1,1),$ $(17,1,1),$ $(18,0,0)$ . for $U$ . For each irreducible root lattice $R_{i}=$ We use the standard basis of roots of the rank we use its standard basis the maximal with the corresponding Dynkin diagram. We denote by . corresponding to this basis. root of R$ where $R$ is the sum of irreducible root lattices $S=U\oplus above, For of we take $H=c_{1}$ and which consists of standard bases and and $e=-c_{1}+c_{2}$ , and one additional consists of Then is shown on the diagram element if $S=U\oplus E_{8}\oplus E_{7}$ . The element $N=49$ of Table 1 as the right-most vertex. It can be easily computed using prescribed by this diagram for basis elements of given pairings above. We obtained the remaining diagrams of cases $N=34,40,41,49,50$ of Table 1. $U\oplus E_{8},$ $U\oplus E_{8}\oplus A_{1},$ $U\oplus E_{8}\oplus E_{7},$ $U\oplus E_{8}\oplus E_{8}.$

$c_{1},$

$A_{1},$

$E_{7},$

$c_{2}$

$r_{1}^{(i)},$

$E_{8}$

$r_{t_{i}}^{(i)}$

$t_{i}$

$\ldots,$

$r_{\max}^{(i)}$

$R_{n}$

$R_{i}$

$r_{1}^{(i)},$

$P_{0}$

$r_{t_{i}}^{(i)}$

$R_{\dot{\tau}}$

$\ldots,$

$r_{0}^{(i)}=c_{1}-r_{\max}^{(i)}.$ $P(\mathcal{M}^{(2,4)})$

$P_{0}$

$\gamma_{1}\in S$

$\gamma_{1}$

$\xi_{i}$

$(\gamma_{1}, \xi_{i})$

$S$

above. To prove finiteness Finiteness of volume of polyhedra defined by the subsets of volume of the polyhedra calculated above with the corresponding diagrams $\Gamma=\Gamma(P)$ of Table 1, one can use methods developed by Vinberg in [Vin72]. $\mathcal{M}^{(2,4)}$

$\mathcal{M}^{(2,4)}$

$S$

$ P=P(\mathcal{M}^{(2,4)})\subset$

133

A.4. CALCULATIONS OF FUNDAMENTAL CHAMBERS

We remind that a subset $T\subset P$ is called elliptic, parabolic, hyperbolic, if its Gram matrix is respectively negative definite, semi-negative definite, hyperbolic. $A$ hyperbolic subset $T$ is called Lann\’er if each its proper subset is elliptic. Dynkin diagrams of all Lann\’er subsets are classified by Lann\’er, e.g. see Table 3 in [Vin72]. They have at most 5 elements. We exclude trivial cases $N=1,2,3,11$ when rk $S\leq 2$ . In all other cases, from our calculations, it easily follows that $P$ generates , and is connected. Moreover, by the classification of affine Dynkin diagrams, one can check that all connected components (for its Dynkin diagram) of any maximal parabolic subset $T\subset P$ are also parabolic, and sum of their ranks is rk $S-2$ . We remind that the rank of a connected parabolic subset $T\subset P$ is equal to $\# T-1.$ From the classification of Lann\’er subsets, it easily follows that the graph has no Lann\’er subgraphs if $N\neq 45,47$ . By Proposition 1 in [Vin72] , then has finite volume. Assume that $N=45$ or $N=47$ . Then the only Lann\’er subset $L\subset P$ consists of two elements defining the broken edge (it is the only one) of Finiteness of volume of . Here is then equivalent to $S\otimes \mathbb{Q}$

$\Gamma(P)$

$\Gamma$

$)$

$\mathcal{M}^{(2,4)}$

$\Gamma.$

$ L^{\perp}\cap \mathcal{M}^{(2,4)}=\emptyset$

$\mathcal{M}^{(2,4)}$

(95)

$\overline{\mathcal{M}^{(2,4)}}=\{x\in S\otimes \mathbb{R}|(x, P(\mathcal{M}))\geq 0\}/\mathbb{R}^{+}$

is the natural extension of . Let $K\subset P$ consists of all elements which are orthogonal to . Looking at the diagrams in Table 1, one can see that $K$ is elliptic and has rk $S-2$ elements. By Proposition 2 in [Vin72], it is enough to show that (it then implies ). Since $\# K=$ rk $S-2$ , the that is the P$ $f_{1}\in (1-dimensional) . There are two more elements edge of and $f_{2}\in P$ such that $K_{1}=K\cup\{f_{1}\}$ and $K_{2}=K\cup\{f_{2}\}$ are elliptic. It follows that the edge and of telminates in two vertices which are orthogonal to and respectively. Any element ) then has . It follows because is a hyperbolic subset. It follows that . Thus, has finite volume for $N=45,47$ either. $\mathcal{M}^{(2,4)}$

$L$

$\Gamma$

$(L\cup K)^{\perp}\cap \mathcal{M}^{(2,4)}=\emptyset$

$K^{\perp}\cap \mathcal{M}^{(2,4)}$

$ L^{\perp}\cap \mathcal{M}^{(2,4)}=\emptyset$

$\mathcal{M}^{(2,4)}$

$r_{1}$

$A_{1}$

$r_{1}$

$K_{1}$

$x^{2}\geq 0$

$L$

$L$

$\neq 0$

$\mathcal{M}^{(2,4)}$

$(L\cup K)^{\perp}\cap \mathcal{M}^{(2,4)}=\emptyset$

A.4.2. Fundamental chambers $h,$

$v_{1},$

$\lambda 4_{+}^{(2,4)}$

$\ldots,$

$v_{6}$

$\mathcal{M}^{(2,4)}$

$\mathbb{R}^{+}x\in r_{1}$

$K_{2}$

$t\underline{hat(x},$

We use orthogonal basis

$A_{2}$

of

of cases $S\otimes \mathbb{Q}$

$N=7$ (Table 2).

where

$h^{2}=8,$

$v_{1}^{2}=\cdots=$

$v_{6}^{2}=-2.$

Case

$7a$

.

As

$P(\mathcal{M}^{(2,4)})$

,

we can take

$f_{3}=v_{3}-v_{4},$ $f_{4}=v_{4}-v_{5},$ $f_{5}=v_{4}+v_{5}$

$f_{1}=v_{1}-v_{2},$ $f_{2}=v_{2}-v_{3},$

with square

$(-4)$

defining the root

134

APPENDIX

system , and $e=(-v_{1}-v_{2}-v_{3}-v_{4}+v_{5})/2+h/4$ with square $(-2)$ . They define the diagram $7a$ . The Weyl group $W=W(D_{5})$ (generated by ) is the semi-direct product of permutations of reflections in $1\leq i\leq 5$ , where $\prod_{i}(\pm)_{i}1=1,$ $1\leq i\leq 5$ , and linear maps see [Bou68]. It follows that $W(e)$ consists of $D_{5}$

$f_{1},$

$f_{5}$

$v_{i},$

$\ldots,$

$v_{i}\rightarrow(\pm)_{i}v_{i},$

$e_{i_{1}i_{2}\ldots i_{k}}=(\pm v_{1}\pm v_{2}\pm\cdots\pm v_{5})/2+h/4$

show where the signs $(-)$ are placed, Omod2 (their number is 16 which is the number of exceptional curves on non-singular del Pezzo surface of degree 4), e.g. we have $e=$ where and

$1\leq i_{1}