Lectures on Curves on an Algebraic Surface. (AM-59), Volume 59 9781400882069

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Annals of Mathematics Studies Number 59

A N N A L S OF M A T H E M A T IC S STU D IE S

Edited by Robert C. Gunning, John C. Moore, and Marston Morse 1. Algebraic Theory of Numbers,

by

H erm ann W eyl

by

3. Consistency of the Continuum Hypothesis,

and N.

B o g o l iu b o f f

edited by S.

21. Functional Operators, Vol. 1,

by

by N.

Ku r t G o d e l

20. Contributions to the Theory of Nonlinear Oscillations, Vol. I,

11. Introduction to Nonlinear Mechanics,

Kr y lo ff

Jo h n v o n N e u m a n n

edited by H. W .

24. Contributions to the Theory of Games, Vol. I,

edited by A.

25. Contributions to Fourier Analysis,

A. P. C a l d e r o n ,

and S.

edited by H. W .

30. Contributions to the Theory of Riemann Surfaces,

edited by L.

33. Contributions to the Theory of Partial Differential Equations,

and F.

T ucker M o r se ,

and A. W .

T ucker

Ku h n

A hlfors

et al .

edited by L.

B e rs , S. B o c h ­

Jo h n

34. Automata Studies,

edited by C. E.

Sh a n n o n

and J.

M cC arthy

edited by H. W .

38. Linear Inequalities and Related Systems,

and A. W .

Ku h n

edited by M.

39. Contributions to the Theory of Games, Vol. Ill,

and P.

and A. W .

T r a n s u e , M.

Ku h n

Zyg m u n d , W .

Bochner

28. Contributions to the Theory of Games, Vol. II,

ner,

L efschetz

T ucker

D r e s h e r , A. W . T u c k e r

W o lfe

40. Contributions to the Theory of Games, Vol. IV,

edited by R.

D uncan L uce

and A. W .

edited by S.

L efschetz

T ucker 41. Contributions to the Theory of Nonlinear Oscillations, Vol. IV, 42. Lectures on Fourier Integrals,

by S.

B ochner

43. Ramification Theoretic Methods in Algebraic Geometry,

by H.

44. Stationary Processes and Prediction Theory,

by S.

F urstenberg

45. Contributions to the Theory of Nonlinear Oscillations, Vol. V, Sa l l e ,

and

Abhyankar

edited by L.

C e s a r i , J. L a ­

S. L e f s c h e t z

by A.

46. Seminar on Transformation Groups, 47. Theory of Formal Systems,

by R.

48. Lectures on Modular Forms,

B orel

et al.

Sm u l l y a n

by R. C.

G u n n in g

49.

Composition Methods in Homotopy Groups of Spheres,

50.

Cohomology Operations,

lectures by N. E.

St e e n r o d ,

by H.

T oda

written and revised by D. B. A.

E p s t e in 51. Morse Theory,

by J. W .

M il n o r

edited by M.

52.

Advances in Game Theory,

53.

Flows on Homogeneous Spaces,

by L.

54. Elementary Differential Topology, 55. Degrees of Unsolvability, 56. Knot Groups,

by L. P.

by G. E.

D r e s h e r , L. Sh a p l e y ,

and A. W . T u c k e r

A u s l a n d e r , L. G r e e n , F. H a h n ,

by J. R.

et al.

M unkres

Sacks

N e u w ir t h

57. Seminar on the Atiyah-Singer Index Theorem, 58. Continuous Model Theory,

by C. C.

Chang

59. Lectures on Curves on an Algebraic Surface, 60. Topology Seminar, Wisconsin, 1965,

by R. S. P a l a i s and H. J. K e is l e r by

D a v id M u m f o r d

edited by R. H.

B in g

and R. J.

B ean

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

BY

David Mumford WITH A SECTION BY

G. M. Bergman

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

1966

Copyright ©

1966, by Princeton University Press All Rights Reserved L. C. Card: 66-17705

Printed in the United States of America

DEDICATION The c o n tr ib u to r s t o t h is volume d e d ic a te t h e ir work t o the memory o f M. K. P o r t , J r . whose warmth and good w i l l have been f e l t by the e n t ir e m ath em atical community.

INTRODUCTION These n o te s a re b e in g p r in te d in e x a c t ly the form in which th ey were f i r s t w r it t e n and d is t r ib u t e d : and w orkin g out my o r a l le c t u r e s . ask a l o t o f the r e a d e r .

supplem enting

In the words o f the e x - e d it o r o f a w ell-kn ow n

jo u r n a l th ey a re w r it t e n i n a s t y l e betw een c lo s e f r i e n d s . "

as c la s s n o te s ,

As such, th ey a re f a r from p o lis h e d and "seldom seen e x c e p t i n p e rs o n a l l e t t e r s

Be th a t as i t may, my hope i s

th a t a w e l l - i n t e n ­

tio n e d re a d e r w i l l s t i l l be a b le to p e n e tr a te th ese n o te s and le a r n some­ th in g o f the b e a u t if u l geom etry on an a lg e b r a ic s u rfa c e . I t was e x p e c te d , when th ese n o te s were w r it t e n , th a t the re a d e r had the f o l l o w i n g background:

he had taken a gradu ate course in

commutative a lg e b r a , he had stu d ie d some A lg e b r a ic Geometry and, in p a r t i c ­ u la r , he had some acqu ain tan ce w ith the th e o ry o f c u rves, and the th e o ry o f schemes, and o f t h e i r cohom ology ( e . g . , D ieu d on n e's Maryland and M on treal L e c tu re N o t e s ).

N o n e th e le s s , b o th t o f i x

id e a s , and to p rove some s p e c ia l­

iz e d r e s u lt s th a t a re needed l a t e r , L e c tu re s 3 - 1 0 a re d evo ted to a q u ick and r a th e r b r e e z y d ig r e s s io n in t o the g e n e r a l th e o ry o f schemes. summarizes what we need from the th e o ry o f cu rv e s .

L e c tu re

11

I a p o lo g iz e to any

r e a d e r who, hopin g th a t he would f in d here in th ese 60 odd pages an ea sy and c o n c is e in t r o d u c t io n t o schemes, in s te a d became h o p e le s s ly l o s t in a maze o f unproven a s s e r t io n s and u n developed s u g g e s tio n s .

From L e c tu re

12

on , we have p roven e v e r y th in g th a t we n e e d . The g o a l o f th ese le c t u r e s i s a com plete c l a r i f i c a t i o n o f one "th eorem " on A lg e b r a ic s u r fa c e s :

the s o - c a lle d com pleteness o f the ch arac­

te r is tic

lin e a r system o f a good com plete a lg e b r a ic system o f c u rves, on a

s u rfa c e

F.

P o in c a re

( c f . R e fe r e n c e s )

If

the c h a r a c t e r is t ic i s in

0 , t h is theorem was f i r s t proven by

1910 by a n a ly t ic m ethods.

U n t il about i 960 ,

no a lg e b r a ic p r o o f o f t h is p u r e ly a lg e b r a ic theorem was known.* Igu sa had shown th a t the theorem , as s ta te d , was f a l s e

thus making the theorem appear even more a n a ly t ic in n a tu re .

1 9 60 , a t r u l y am azing developm ent o ccu rred :

In 1955,

in c h a r a c t e r is t ic

p

But about

in the course o f w ork in g out

the m aster p la n th a t he had l a i d out f o r A lg e b r a ic Geometry—in c o r p o r a tin g some o f the key id e a s o f K o d a ir a 's and S p e n c e r's d e fo rm a tio n th e o ry —G rothend ie c k had o c c a s io n to w r it e out some o f the C o r o lla r ie s o f h is th e o ry ( c f . h is Bourbaki expose 2 2 1 , pp. 2 3 - 2 ^ ) . P u ttin g h is r e s u lt s to g e th e r w ith a * A lth ou gh an e n d le s s and d e p re s s in g c o n tr o v e r s y obscured t h is f a c t . v ii

v iii

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

r e s u lt o f C a r t ie r —th a t group schemes in c h a r a c t e r is t ic

0

fin d s th a t t h is o ld problem has been c o m p le te ly s o lv e d : b r a ic p r o o f i s a v a ila b le in c h a r a c t e r is t ic

0,

b ) a l l the m achinery i s

rea d y a t hand f o r o b ta in in g , in c h a r a c t e r is t ic c o n d itio n s f o r the v a l i d i t y o f the theorem . p o in t w hich the I t a l i a n s had overlook ed ?

a re reduced—one a) a p u r e ly a l g e ­

p, n e cessa ry and s u f f i c i e n t

What was the key, the e s s e n t ia l

There i s no doubt a t a l l th a t i t

i s the sy s te m a tic use o f n ilp o t e n t elem en ts:

in p a r t ic u la r , a s y s te m a tic

a n a ly s is o f a l l fa m ilie s o f curves on a su rfa c e o v e r a param eter space w ith o n ly one p o in t , bu t w ith n o n - t r i v i a l n ilp o t e n t s tru c tu re s h e a f. ia n s had, in

The I t a l ­

a sen se, done t h i s , but o n ly when the r in g o f fu n c tio n s on the

base was S tu d y 's r in g

o f du al numbers

k [ e ] / ( e 2) .

in g a t f i r s t - o rd er d efo rm a tio n s o f a cu rve.

But

T h is i s the same as lo o k ­ th ey ig n o re d h ig h e r o rd e r

n ilp o t e n t s and h ig h e r o rd er d e fo rm a tio n s . The o u t lin e o f th ese le c t u r e s i s as f o llo w s —L e c tu re s 1 and 2 g iv e an i n t u i t i v e in tr o d u c tio n t o the problem and p re s e n t in o u tlin e 2 an­ a l y t i c p r o o fs .

L e c tu re s 3 through 10 r e c a l l b a s ic n o tio n s about schemes.

L e c tu re s 11 through 21 d e a l w ith b a s ic q u e s tio n s on the th e o ry o f s u rfa c e s . In p a r t ic u la r ,

th ey g iv e a c o n s tru c tio n o f u n iv e r s a l fa m ilie s o f cu rves on

a s u rfa c e —the s o - c a lle d H ilb e r t scheme; and o f u n iv e r s a l f a m ilie s o f d i v i s o r c la s s e s on a s u rfa c e —the s o - c a lle d P ic a rd scheme.

L e c tu re s 22 through 27

d e a l w ith the a p p lic a t io n o f the whole th e o ry t o the main problem :

th ese

in c lu d e a lo n g le c t u r e by G. Bergman g i v i n g a s e lf- c o n t a in e d d e s c r ip t io n o f the W it t r i n g schemes. I would

lik e

t o c a l l a t t e n t io n t o s e v e r a l g e n e r a liz a t io n s and

a p p lic a t io n s o f our r e s u lt s which were o m itted so as to

g e t d i r e c t l y t o the

main r e s u l t . a)

The method by which we have co n s tru c te d the u n iv e r s a l

fa m ily o f cu rves on a s u rfa ce

F

g iv e s w ith o u t any change a c o n s tr u c tio n o f

the u n iv e r s a l f l a t fa m ily o f subschemes o f any scheme n o e th e ria n

S, i . e . ,

o f the H ilb e r t scheme.

X, p r o j e c t i v e o v e r a

In p a r t ic u la r ,

the e x p l i c i t

e s tim a te s ob ta in ed in L e c tu re 1^ en ab le ont t o c a r r y through t h is c o n s tru c tio n -w h ic h i s G ro th e n d ie c k 's o r i g i n a l cons tru e tion-^wi thou t the i n d i r e c t a r ­ guments u s in g the con cept o f " l im it e d f a m i l i e s " which he used ( c f .

h is

"F o n d em en ts"). b) o f a s u rfa ce scheme over

F

The method by which we have co n s tru c te d the P ic a r d scheme g e n e r a liz e s so as t o c o n s tru c t the P ic a r d scheme o f any

X, p r o j e c t i v e and f l a t o v e r a n o e th e ria n S

S, whose g e o m e tric f i b r e s

a re reduced and connected and such th a t the components o f i t s a c ­

tu a l fib r e s over

S

are a b s o lu t e ly i r r e d u c ib le .

T h is c o n s tr u c tio n i s r e ­

la t e d to the one I o u tlin e d a t the I n t e r n a t io n a l Congress o f 1 9 6 2 , and t i e s up w ith the methods used in Chapters 3 and 7 o f my book T h e o ry .

G eom etric In v a r ia n t

INTRODUCTION c)

ix

One can use the r e s u lt s o f L e c tu re 18 t o g iv e a v e r y easy

p r o o f o f the Riemann H yp oth esis f o r cu rves o v e r f i n i t e p r o o f o f M attu ck -T ate ( c f .

R e fe r e n c e s ).

fie ld s .

T h is i s

the

I f you have rea d through L e c tu re 1 8 ,

and know the fo r m u la tio n o f the Riemann H yp oth esis v i a the F roben iu s mor­ phism, you can rea d t h e i r paper w ith o u t d i f f i c u l t y and you should.

Cambridge March,

1966

CONTENTS INTRODUCTION..................................................................................................................... LECTURES

1: 2:

Raw M a te r ia l on C urves on S u r fa c e s , and the .................................................................... Problem s S u gg este d The Fundam ental E x is t e n c e Problem and Two A n a ly t ic

3:

P ro o fs ............................................................................................... P re-schem es and t h e i r A s s o c ia te d "F u n cto r o f P o in ts '

b:

Uses o f th e F u n cto r o f P o in ts .............................................

Appendix to L e c tu re b:

1 7 11 17

Re R e p r e s e n ta b le F u n c to rs and Z a r i s k i

5:

Tangent Spaces ............................................................................. P ro j and I n v e r t i b l e Sheaves ..................................................

6: 7:

P r o p e r tie s o f Morphisms and Sheaves ................................ Resume o f the Cohomology o f C oheren t Sheaves on Pn

8: 9: 10:

F l a t t e n in g S t r a t i f i c a t i o n s .................................................. C a r t i e r D iv is o r s ........................................................................ F u n c t o r ia l P r o p e r t ie s o f E f f e c t i v e C a r t i e r D iv is o r s

11: 12:

B ack to the C l a s s i c a l C a s e .................................................. The O v e r - a l l C l a s s i f i c a t i o n o f C urves on S u rfa c e s .

13:

L in e a r System s and E x a m p le s ..................................................

1I4-:

Some V a n is h in g T h e o r e m s ..........................................................

15: 16: 17: 18: 1 9: 20: 21: 22: 23:

U n iv e r s a l F a m ilie s o f C urves ............................................. The Method o f Chow S c h e m e s .................................................. Good Curve s ...................................................................................... The Index Theorem ........................................................................ The P ic a r d Scheme: O u t l i n e .................................................. Independent 0 -c y c l e s on a S u rfa c e .................................... The P ic a r d Scheme: C o n clu sio n ............................................. The C h a r a c t e r i s t i c Map o f a F a m ily o f C urves . . . The Fundam ental Theorem V ia K o d a ira -S p en ce r . . . .

2b :

The S tr u c tu r e o f 0 .................................................................... The Fundam ental Theorem V ia G r o th e n d ie c k - C a r tie r . . R in g Schemes: The W itt Scheme, hy G.M. Bergman

25: 26:

v ii

25 27 37 bl

57 61 69 75 83 91 99 105

111 119 1 27

133 139 U5 1 51 1 57 1 61 167 1 71

Appendix to L e c tu re 26: .......................................................................................... 27: The Fundam ental Theorem in C h a r a c t e r i s t i c p . . .

1 89

BIBLIOGRAPHY-.....................................................................................................................

199

xi

193

LECTURE 1 RAW MATERIAL ON CURVES ON SURFACES, AND THE PROBLEMS SUGGESTED We s h a ll be concerned e n t i r e l y w ith a lg e b r a ic geom etry o v e r a f i x e d a l g e b r a i c a l l y c lo s e d f i e l d purpose i s

k

( o f a r b it r a r y c h a r a c t e r i s t i c ) .

p r o je c t iv e over

k, and, in p a r t ic u la r ,

the f a m ilie s o f cu rves

By a curve we mean e it h e r a f i n i t e a l s u b v a r ie t ie s o f c f.

Example 1 :

F.

C

Z

C

F,

on

F.

1 -dim en sion ­

or a sh eaf o f

[These a re e q u iv a le n t con cep ts—f o r p r e c is e d e f i n i ­

L e c tu re 9 . ] F = p2.

Then, as i s w ell-k n ow n , e v e r y curve

d e fin e d by a homogeneous form ta ch t o

sum o f i r r e d u c ib le ,

F, w ith p o s i t i v e m u l t i p l i c i t y :

p r in c ip a l id e a ls on t io n s ,

Our c h i e f

to study th e geom etry on a n o n -s in g u la r a lg e b r a ic s u rfa c e

its

d egree

curves o f d e g re e up to s c a la r s :

d i.e .,

F (x Q, x 1, x 2) .

d, i . e . ,

the d e g re e o f

F,

on

p2

is

one can a t ­

and the fa m ily o f a l l

i s p a ra m e trize d by the s e t o f a l l by a p r o j e c t i v e

C

In p a r t ic u la r , F

o f d eg ree

d,

Then e v e r y curve

C

space o f dim ension

4- 2 ) _

(d + i ) ( d 2

Example 2 : on

F

F = P 1 x P 1

(i.e .,

a q u a d ric in

p ^ ).

i s d e fin e d by a bi-hom ogeneous form F (x 0 , x 1 j y Q, y 1)

w ith two d e g re e s

d

and

e.

d

and

e

can be in t e r p r e t e d as the d e g re e s

o f the c o v e r in g s p, , p 2 : g iv e n by the two p r o je c t io n s o f and

e,

C — P,

P1 x P1

onto

P1 .

A g a in , f o r e v e r y

d

th e re i s a s in g le fa m ily o f cu rves p a ra m etrized by a p r o j e c t i v e

space, t h is tim e o f dim ension; (d + 1 ) (e + 1 ) - 1 The phenomenon o f the l a s t two exam ples can be g e n e r a liz e d by the con cept o f a lin e a r system . as u su a l,

(f)

If

f

i s an a lg e b r a ic fu n c tio n on

stand f o r the fo rm a l sum: 1

F,

le t,

2

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE E

°rd E ( f )

• E

a l l 1-d im en sio n a l i r r e d u c ib le s u b v a r ie t ie s E where

o r d g (f)

is

the o rd er o f the z e ro or p o le o f

s o c ia te d t o any curve o n ly a t

C

f

at

E.

Then a s ­

one has the v e c t o r space o f fu n c tio n s w ith p o le s

C: £ (C )

= Cf|

nieans a l l

(f)

+ C > 0}

(H ere

E nj_Ei > 0

£ (C ),

one then can d e fin e the f o llo w in g fa m ily o f

C'

> 0 .)

Ca

Ca

S in ce

-

If

f Q, . . . , f n

(2

♦ C

o n ly depends on the r a t i o s o f the

a^,

fa m ily o f cu rves p a ra m etrized by a p r o j e c t i v e dim

a re a b a s is o f

c u rves, w hich co n ta in s

£ (C )

t h is i s an ir r e d u c ib le

space o f dim ension:

- 1

L in e a r systems a re the s im p le s t f a m ilie s o f cu rves on a su rfa c e

F

and

the o n ly typ e o c c u rrin g in Examples 1 and 2. D e fin itio n :

Two cu rves C1

and

C2 a re l i n e a r l y e q u iv a le n t

if

e q u iv a ­

le n t ly : i)

3

ii)

a fu n c tio n

C1, C2

We w r it e Example 3:

f

a re i n

on

F such th a t

(f)

= C1- C2, or

the same lin e a r system.

C1 = C2 f o r t h is con cep t.

Let

g

be an

A g a in , g iv e n a curve

C

e l l i p t i c curve ( o v e r on

k ) , and

le t

F= P 1

F, we can a s s ig n t o Ctwo d e g re e s

x &.

d and

e,

as the o rd e rs o f the c o v e rin g s C —►P1 5 o b ta in ed by p r o je c t in g .

Both

d

C— &

> 0 and

e > 0

and e it h e r

d > 0

or

e > 0. Case

i)

form

a s in g le e -d im e n s io n a l l in e a r system .

d = 0:

Then C

Case

ii)

d > 0:

is

o f the form

The s e t o f a l l

C

E ? _ 1P. x

o f typ e (d , e )

6 , and a l l

form s

th ese

C

ani r r e d u c ib le

d (e + 1 ) -d im en sio n a l fa m ily o f cu rv e s , but i t i s n o t a l in e a r system . R ath er i t

i s fib r e d by

d (e + 1 )- 1 -d im en sio n a l l in e a r s u b fa m ilie s .

D e fin itio n :

Two cu rves C1, C2

C2

con ta in ed in one fa m ily o f curves p a ra m etrized by a connected

a re b o th

a re a l g e b r a i c a l l y e q u iv a le n t i f

C1

and

v a r ie ty . W ith t h is te rm in o lo g y , we can say th a t on l in e a r e q u iv a le n c e d i f f e r . form u la i n Case i i ) Case i )

when

P1 x g,

A nother p o in t to n o t ic e i s

a lg e b r a ic and

th a t the dim ension

does n ot s p e c ia liz e t o the d im en sion a l form u la in

d = 0:

t h is i s

the phenomenon o f superabundance.

3

RAW MATERIAL, AND THE PROBLEMS SUGOISTED Example in g o f

7

Let P1

be a " g e n e r ic " curve o f genus 2 , i . e . ,

n a tes o v e r th e prim e f i e l d

7.

It

a double c o v e r ­

branched a t s ix p o in ts w ith in dependent tra n s c e n d e n ta l c o o r d i­ ch ar. ^ 2 ) .

(if

Let

P

be the ja c o b ia n o f

R e c a ll th a t (1)

P

i s a n o n -s in g u la r a lg e b r a ic s u r fa c e ,

( 2)

P

i s a ls o an a lg e b r a ic grou p,

(3 )

i n a n a tu r a l way,

turns out th a t e v e r y curve

curve

d r,

y

it s e lf

i s a curve on

C

on

i s a l g e b r a i c a l l y e q u iv a le n t t o a

P

f o r a s u it a b le p o s i t i v e i n t e g e r

e q u iv a le n t t o a s u it a b le t r a n s la t io n o f group s t r u c t u r e ). d

M oreover,

dr ( i n the sense

C is lin e a r ly o f the g iv e n

The s e t o f a l l cu rves a l g e b r a i c a l l y e q u iv a le n t t o

i s an ir r e d u c ib le fa m ily o f dim ension have dim ension

d.

P.

2

- 1.

In f a c t ,

d

2

+ 1,

and i t s

one can d e fin e a map:

a l l cu rves a l g . e q u iv a le n t t o

dr

l in e a r e q u iv a le n c e where a «-*■ image o f

dr

lin e a r s u b -fa m ilie s

dr ' under t r a n s la t io n by

a.

J In f a c t ,

t h is map

f a c t o r s as f o llo w s : p m u lt, b y

d ^

cu rves a l g . e q u iv a le n t to dr

p b is e c t io n ^

l in e a r e q u iv a le n c e T h is in d ic a t e s a g e n e r a l p o in t : lin e a r e q u iv a le n c e ],

th e s e t [a lg e b r a ic e q u iv a le n c e modulo

tends t o be in depen den t o f th e fa m ily o f cu rves con­

s id e r e d . One should c o n tr a s t t h is s u rfa c e K:

t h is i s d e fin e d

^ 2).

s e x t ic cu rve (c h a r . d • h,

(a )

h

w ith i t s P2

"Kummer" co u n te rp a rt

branched in a g e n e r ic

Here a l l cu rves a re l i n e a r l y

e q u iv a le n t to

the in v e r s e image o f a l i n e in p Q, and th e dimen2 s io n o f t h is fa m ily i s d + 1 (a s a b o v e ). I t i s s im ila r t o F a ls o in

th a t

where

P

as th e dou ble c o v e r in g o f

is

( y 2) = 2

on

P,

(h 2) = 2

t io n —c f . L e c tu re 1 2 ] , and (b )

b o th P

w ith n e it h e r z e ro s n or p o le s .

T h is

s u rfa c e s in

on

K [ (D2) d en otes s e l f - i n t e r s e c ­

and K

is

K

adm it double d i f f e r e n t i a l s

o f the same typ e as the q u a r tic

p^.

In f a c t , we have touched b r i e f l y on e v e r y c la s s o f a lg e b r a ic su r­ fa c e s a d m ittin g a double d i f f e r e n t i a l w ith no z e ro s ( i . e . , c a l lin e a r system ) :

an a n t i- c a n o n i­

f o r rea son s stemming from S e rre d u a lit y ,

on th ese s u rfa c e s i s p a r t i c u l a r l y sim p le.

the geom etry

To b r in g out some fu r t h e r f e a ­

tu re s o f s u r fa c e s , we w i l l d is c u s s an oth er r a t i o n a l s u rfa c e : Example 5 : P2

in

Let

P

th e r a t i o n a l cu rves Let on

H be th e l i n e P

p1 x P ^ .

w hich a re the in v e r s e im ages o f in

P2

from P 1

to

P2,

C

on

F,

E1

Let P1

and l e t

w hich i s th e c lo s u r e o f the in v e r s e image o f

t o e v e r y cu rve

P1 ,

be th e s u rfa c e o b ta in ed by b lo w in g up two p o in ts

p 2 [o r b y b lo w in g up one p o in t i n

and D

and P2

on

P.

be th e curve

£ - P 1 - P2 .

one can a tta c h th re e c h a ra c te rs

E 2 be

Then

k1, k2, and

I,

k

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

where k1, k2

and

A

a re n o n -n e g a tiv e and n ot a l l z e r o ;

a l l cu rves w ith c h a ra c te rs

k1, k2, A

and the s e t o f

form the s in g le lin e a r system con­

t a in in g k1E l + k2E2 + AD But u n lik e the s it u a t io n on

P 1 x P.,,

n o t a l l th ese systems a re "good "

systems o f cu rves. Case i )

£ > k2

I f A > k^,

cu rves

E1,

E2,or

c o n ta in in g

D

and

k1 + k2 > £,

then none o f the th re e

i s a component o f a l l cu rves in the lin e a r system

k1E1 + k2E2 + AD,

and t h is lin e a r system has the p r e d ic t a b le

dim ension: (A+1) ( £+2) --------g--

(* ) Case i i )

If E1 ,

cu rves

A

< ^ ,

E2, or

D

( A - k . ) (A -k . +1) ( A -k2) (A-kp+1) ^ ---------------------------------------

£ < k2,

or

k1 + k2 < £,

i s a component o f a l l the

1

then one o f the th re e cu rves in q u e s tio n , and,

in g e n e r a l, t h is fa m ily i s a ls o superabundant, i . e . ,

i t s dim ension i s

b ig g e r than th a t p r e d ic te d by ( * ) . Another way o f t e l l i n g the "g o o d " from the "bad" systems o f cu rves is

t h is : the system o f cu rves l i n e a r l y e q u iv a le n t t o k1El + k2E2 + AD

A > k1

i s the fa m ily o f h yp erp lan e s e c tio n s o f F f o r some embedding o f F in PN

A > k2 k1 + k2 > A

k1E 1 + k 2E2 +

Here th e c o n d itio n on the l e f t d e fin e s the n o tio n : AD

i s v e r y am ple. W ith a l l t h is d ata b e fo r e us, what q u e s tio n s emerge as th e n a tu r a l

ones t o pose in s tu d y in g the cu rves on a g e n e r a l su rfa c e

F ?

I th in k

fo u r b a s ic l in e s o f study a re su ggested : (i)

the problem o f Riemann-Roch:

G iven a curve

C,

t o d eterm in e the dim ension o f the lin e a r system o f curves c o n ta in in g

C.

We s h a ll see below th a t t h is i s e q u iv a le n t

t o the problem o f computing dim H° ( £ ) where Op

£

i s a s h ea f on

F , l o c a l l y isom orp h ic t o the sh ea f

o f r e g u la r fu n c tio n s . (ii)

the problem o f P ic a r d :

To d e s c r ib e the fa m ily o f

a l l a lg e b r a ic d efo rm a tio n s o f a curve s u b fa m ilie s . of

C,

if

C

modulo i t s

lin e a r

I t turns out th a t t h is q u o tie n t i s independent C

i s good, and t h is q u o tie n t le a d s t o the P ic a r d

scheme and/or v a r i e t y .

RAW MATERIAL, AND THE PROBLEMS SUGGESTED

(iii)

Good v s . Bad cu rv e s :

One can ask when i s

C

What makes

v e r y am ple,

C

when i s

good and bad? C

abundant, what a re the r e a l l y bad " e x c e p t io n a l" p la y the r o l e o f

E1 , Eg

and

D

super­ C

w hich

i n Example 5 above?

P a r t i c u l a r l y s i g n i f i c a n t i s th e q u e s tio n o f th e " r e g u l a r i t y o f the a d jo in t " (iv )

(= " K o d a ir a 's v a n is h in g th eorem ") c f .

the components o f th e s e t o f a l l cu rves

E s p e c ia lly , what f i n i t e n e s s

C

statem en ts can be made?

L e c tu re 1 on

P_:

Ex­

amples a re the theorem o f the base o f Neron and S e v e r i, and the theorem th a t o n ly a f i n i t e number o f components r e p r e s e n t cu rves o f any g iv e n d e g re e .

LECTURE 2 THE FUNDAMENTAL EXISTENCE PROBLEM AND TWO ANALYTIC PROOFS We s h a l l a n a ly z e problem i i )

more c l o s e l y .

The r e a l n a tu re o f th e

problem becomes c le a r e r when one p a s s e s from c u rv e s to d i v i s o r s . By a d i v i s o r on F we mean e i t h e r a f i n i t e sum o f i r r e d u c i b l e , 1 -d im en sio n a l s u b v a r i e t i e s , w ith ( p o s i t i v e or n e g a tiv e ) m u l t i p l i c i t y : Z nj_C-p nj_ € Z, or a s h e a f o f f r a c t i o n a l i d e a l s , i . e . , a c o h eren t su b sh ea f o f th e c o n s ta n t sh e a f

K: K(U) = fu n c tio n f i e l d k ( F ) , a l l

U

( c f . L e c tu re 9 f o r p r e c i s e d e f i n i t i o n s ) . The s e t o f a l l d i v i s o r s on form s a grou p , w hich we d en ote G(F) . P u t:

F

Ga (F) = subgroup o f d i v i s o r s o f th e form C1 - C2 , where C1 , C2 a re a l g e b r a i c a l l y e q u iv a le n t c u r v e s , G^(F) = subgroup o f d i v i s o r s o f th e form

C1 - C2,

where

C1 = C2 , o r , e q u i v a l e n t l y , th e subgroup o f d i v i s o r s o f form ( f ) , f € k ( F ) . Now i f C i s any cu rve on F , and c u rv e s a l g e b r a i c a l l y e q u iv a le n t to

(C^l a € S) i s th e fa m ily o f a l l C = CQ, one can d e fin e a map:

o j moauio l i n e a r ' s u b fa m ilie s b y mapping

a

to th e d i v i s o r

C

- CQ.

One ch eck s im m ed iately th a t i t

i s a lw a y s i n j e c t i v e , and i t can be shown t h a t f o r s u f f i c i e n t l y "good" (?J) c u r v e s , i t i s s u r j e c t i v e . For t h i s r e a s o n , problem ( i i ) becomes in d epen d en t o f C, i n most c a s e s , and a s k s sim ply-w h at i s th e s tr u c tu r e and dim en sion o f th e group

G (F)/G^(F)

i n v a r i a n t l y a tta c h e d to

F ?

A g ain w ith o u t p r o o f s , we would l i k e to m ention th e co h o m o lo g ical i n t e r p r e t a t i o n o f th e s e grou p s: L et o* = sh e a f o f u n it s i n th e s tr u c t u r e s h e a f K* = s h e a f o f u n it s i n K. Then: le a d s to :

o_

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

8

o - H ° (K * )/ k *

-

H °(K ? o *)

?ll

group o f

Ga (F )/ G ^ (F ) F

H1(o * )

-

0

2II

G *(F ) T h e r e fo r e ,

-

G (F) i s a subgroup o f

H1 (o _*), the s o - c a lle d P ic a rd

(d e fin a b le on any r in g e d - s p a c e ).

Now the work o f C asteInuovo and Matsusaka has shown th a t the group G (F )/ G ^ (F )

can be g iv e n , in a n a tu r a l way, the s tru c tu re o f an a lg e b r a ic

group—i n f a c t , is

an a b e lia n v a r i e t y .

the dimension?

The e s s e n t ia l p o in t i s ,

Here we have an e x is te n c e problem :

how ever, what

can we p r e d ic t the

dim ension o f the s e t o f s o lu tio n s o f an e s s e n t i a l l y n o n -lin e a r problem by means o f some l i n e a r d a ta , e . g . ,

the cohomology o f a coh eren t sheaf?

It

was c o n je c tu re d by S e v e r i th a t: (A )

dim Ga (F )/ G ^ (F )

where

£ = s tru c tu re sh ea f on

F,

=

q = p - p ) . T h is g a k = c , and was d is p ro v e n by Igu sa

c h a r(k ) / 0 .

The s im p le s t way to m o tiv a te l e f t i s a subgroup o f

H1( £ * ) ,

kind o f "e x p o n e n tia l" from way i s

to tra n sform

a curve

C

on

H1 (o )

( i n h is lan gu age,

was proven by P o in ca re in 1 9 0 9 , when in 1953, when

dim

F,

(A )

(A ) i s to n ote th a t the term on the

and to guess th a t th e re should be some

H1 (o )

to

H1 ( £ * ) ,

( c f . b e lo w ).

and in t h is form , i t

i s a s p e c ia l case o f th e g e n e r a l

K od aira-S p en cer e x is te n c e problem f o r d e fo rm a tio n s . a g a in th a t

(Ca | ot e S)

To see t h i s ,

i s a fa m ily o f d efo rm a tio n s o f

be the sh ea f o f s e c tio n s o f the normal bundle t o n o n - s in g u la r ).

A second

in t o a statem ent con cern in g the d efo rm a tio n s o f

C

in

C = CQ. F

suppose Let

(assume

N C is

Then th e re i s a fundam ental c h a r a c t e r is t ic map: / Tangent Space \^ \ to S at a = o f

TrOm x M

Roughly sp eak in g, a sm all neighborhood o f t o the norm al bundle t o

C

in

F,

C

in

w h ile a curve

d e fin e s a s e c tio n o f t h is n eigh borh ood:

as

a -► 0

F

i s n e a r ly isom orph ic

Ca ,

fo r

a

t h e r e fo r e ,

n ear

0,

th ese cu rves

can be a s y m p to tic a lly i d e n t i f i e d w ith s e c tio n o f the normal bundle t o in

F.

C

The key e x is te n c e problem i s now:

(B)

f o r s u ita b le

(C ^ },

p

is b ije c tiv e

In c id e n t a lly , in t h is form , the c o n je c tu re can be e q u a lly w e l l posed f o r s u b v a r ie tie s in o th e r v a r i e t i e s o f a r b it r a r y codim ension, e . g . , fo rm a tio n s o f cu rves in

p^.

U n fo rtu n a te ly , i t i s f a l s e even in

fo r de­ ch ar. 0

f o r some p a t h o lo g ic a l space cu rves. To connect c o n je c tu re s sheave s :

(A ) and ( B ) , we use the e x a c t sequence o f

FUNDAMENTAL EXISTENCE PROBLEM: TWO ANALYTIC PROOFS n)

22

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE (* )

if

m C A

A p p lic a t io n 2:

F ib r e s o f a morphism.

pre-sch em es, and y

i s a maximal i d e a l ,

y e Y

then

Let

be any p o in t .

A/ m ^ n.

f : X -► Y

Let

K (y )

QED be a morphism o f

= re s id u e f i e l d

of

oy

d eterm in es a c a n o n ic a l morphism: K ( y ) -----

Spec v ia

/ th e p t . -► y Oy

I K (J)

( c a n o n ic a lly )

One form s the f i b r e p ro d u ct: X X Spec

T h is i s

the scheme-the ore t i c

i s a ge o m e tric

is

p o in t o f

K (y )

fib r e o f

Y,

f .

= f -1 ( y ) ,

or

S im ila r ly , i f

X^.

g: Spec n -*• Y

then the f i b r e p rod u ct:

c a lle d the g e o m e tric f i b r e o f

f

o ver the g iv e n g e o m e tric p o in t .

In

t h is lan gu age, one has the d r o l l r e s u l t : P r o p o s it io n :

Let

k C K

be two

induced by the in c lu s io n o f [K/k

is

k

a group i s

and l e t

f : Spec K -►

Spec k be

K. Then,

s e p a ra b le ] [~ one (and hence a l l ) g e o m e tric f i b r e s Lo f f a re reduced schemes (P r o o f i s

A p p lic a t io n 3:

fie ld s ,

in

l e f t t o r e a d e r .)

D ir e c t d e f i n i t i o n o f a group pre-schem e

sim ply a s e t

X

m u lt: in v e r s e : id e n tity : s a t i s f y i n g w ell-kn ow n r e l a t i o n s . s i s t s in th e fu n c to r morphisms o f fu n c to r s :

/S.

A fte r a l l ,

p lu s th re e maps:

h^. (on the

X x X — X X -► X ( e ) -► X T h e r e fo r e , a group pre-schem e

X/S

c a te g o r y o f pre-schem es /S)p lu s th re e

con

23

USES OP THE FUNCTOR OP POINTS m u lt:

h^. x h^. “ ►h^.

in v e r s e :

h^ -► .h^

id e n tity : {1 e l t . s a t i s f y i n g th e same i d e n t i t i e s . h^. XjgX>

But:

(a )

h^. x h^.

i s isom orp h ic t o

e l t * fu n c'to:i:>^ i s isom orp h ic to

f i n a l o b je c t i n our c a te g o r y . if

fu n c to r ) -►

T h e r e fo r e ,

X

hg,

S

b e in g the

i s a group pre-schem e

/S

one i s g iv e n th re e morphisms: m u lt:

X x X -► X S

in v e r s e :

X

id e n tity :

S -► X

X

s a t is f y in g th e same i d e n t i t i e s . A f i n a l p o in t n o t to be fo r g o t t e n : /S,

fo r a l l

T/S,

the

sense do the o rd in a ry p o in ts o f A p p lic a t io n 1^:

X -► X

X

D e f i n i t i o n o f a scheme. be the i d e n t i t y .

X X

i s a group pre-schem e

form a group:

form a group Let

X

(e v e n i f

but in no

S = Spec ft ).

be a pre-schem e, and l e t

The induced morphism

A = ( 1X , 1X ) : is

if

T -v a lu e d p o in ts o f

X

X x X

c a lle d th e d ia g o n a l.

P r o p o s it io n - D e f in it io n : i) ii)

a (X )

is

X

i s a scheme i f

c lo s e d in

e q u iv a le n t ly :

r

X x X,

f o r e v e r y p a ir o f morphisms

11

Y

X ,

f2

( y € Y| f 1(y ) P r o o f: X x X

ii) on

===> i ) X;

i)

= f 2( y ) )

by ta k in g

===> i i )

i s a c lo s e d subset o f

Y = X x X,

by fa c t o r in g

(■^1 >^ 2^ Y --------------

Y

f^ = i th p r o je c t io n

p^

of

f^ -^1

7 X ,

Xx X P2

and n o tin g th a t (y € Y| f^y = f 2y)

=

(f,, f 2 )- 1 [A(X)]

QED From now on, we w i l l d e a l o n ly w ith schemes, u n less o th e rw is e s p e c if ie d .

APPENDIX TO LECTURE k RE

REPRESENTABLE FUNCTORS AND ZARISKI TANGENT SPACES

As an a p p lic a t io n b o th o f th e con cep ts o f fu n c to r s and o f n ilp o t e n t s , we connect th ese t o the g e o m e tric con cep t o f the Z a r is k i ta n gen t space. Assume th a t

X

i s a scheme

r a t i o n a l p o in t , i . e . , phism

k

over a f i e l d

th e g iv e n

k,

and

homomorphism

th a t

k ox

x e X is a

in du ces

k-

an iso m o r­

K (x ) .

D e fin itio n : 2 t o m/m is

If

m C o„

i s the maximal i d e a l ,

the Z a r is k i ta n gen t space

Tx

then the d u a l v e c t o r to

X

at

space

x.

Now c o n s id e r the i n t e r e s t i n g c la s s o f schemes: D e fin itio n :

If

V

i s a v e c t o r space (a lw a ys f i n i t e d im en sio n a l)

o ver

k,

le t

ly where

k 0 V

i s a r in g v ia

= V

Spec (k 0 V )

2

= (o ) .

,

N ote th a t one has two homomor­

phism s: k CODIM 0 ' THE GENERIC ’ POINT

l STALK Q (X ,/X 0)

CODIM I

SPE SPEC(Z) ve.)

ioj

ioi

in

ipi

101

STALK Z(R) STALK Q E x e r c is e :

What i s

the p o in t ( * ) ?

A more w e ig h ty q u e s tio n i s what a re the S -valu ed p o in ts in i.e .,

what i s

the fu n c to r o f

h

.

pn ,

The answer to t h is q u e s tio n in v o lv e s

n us im m ed ia tely in a new c o n c e p t.: D e fin itio n :

If

X

i s a l o c a l r in g e d space, a sh eaf

such th a t th e re e x is t s a c o v e r in g

{U^}

of

X

£

of

£x -modules

f o r which

£| as is

c a lle d an i n v e r t i b l e

s h e a f.

More c o n c r e t e ly , what i s

0^ ,

p h ic t o

o^-m odules,

such an

the e s s e n t ia l p a r t o f

£

can be co n s tru c te d b y s t a r t in g w ith as sheaves o f

o^-modules on

Horn ' as sheaves o f o^-modules [where U j, o_x )

U^ n U^.

£ ? Since l o c a l l y i t i s in the p a tc h in g : on each

U^,

i s isom or­ i.e .,

£

and p a tc h in g th ese

But

(%'u.

h e Horn correspon ds to

h( 1 ) e rfU ^ n U j,

ox ) ;

and

f e r(U ^ n

correspon ds t o the homomorphism g iv e n by m u lt ip lic a t io n by

f ].

Now d e fin e : D e fin itio n : i) or

ii)

An elem ent

s e r (U ,

Oj)

i s a u n it i f e q u iv a le n t ly :

th e re e x is t s a m u l t i p l i c a t i v e in v e r s e fo r a l l

x e U, the induced elem en t

the maximal i d e a l

s

s " 1 € r (U , Oy.) -X '

in

i s n o t in

29

P ro j AND INVERTIBLE SHEAVES I t is

c le a r from ( i i )

denote

o*.

I t is

m u lt ip lic a t io n .

th a t the u n its form a subsheaf o f

c le a r from ( i )

Now i t

is

th a t

a*

o^. -w hich we w i l l

i s a sh e a f o f groups under

c le a r th a t the isom orphism s o f

w ith i t s e l f

a re: Isamas sheaves o f -module s T h e r e fo r e ,

to c o n s tru c t

i £,

m u lt ip lic a t io n by a u n it

a^.

s^

T h is means th a t

\

elem ent

• sj k

n U^ .

D Uj n U^,

• sk±

n Uj> 2%)

~ u ra ts l n r ( ui

must be patched to i t s e l f on over

tio n s must be com p a tib le on si j

° x lu ,n u J i j

t3

=

1

on

it

ui

n

by

Sin ce a l l th ese i d e n t i f i c a ­ f o llo w s th a t:

n uj n uk •

(s ..) form a 1-C zech c o - c y c le , and we have d e fin e d an 1 1J H (X , oJ£) . The main, but elem en tary, r e s u lt in t h is d i ­

of

r e c t io n i s :

\

P r o p o s it io n 2:

depends o n ly on

betw een th e s e t o f i n v e r t i b l e set

£,

and t h is s e ts up an isom orphism

sheaves on

X-modulo isomorphismr-and th e

H1 (X , o £ ) .

D e fin it io n : Remarks:

P ic (X )

A)

P ic

= H1 (X ,

(X)

a sh e a f o f grou ps.

£1 0 £2 any

X

£ 1 0 £2;

(U ^ ),

is

B) f

and

if

£1

and i f

L2

and

£g

c le a r sin c e

o*

a re i n v e r t i b l e

i s g iv e n by the c o - c y c le

i s g iv e n by

t^ j

s^

is

sheaves, s^

f o r the same c o v e r in g ,

sim p ly the sh ea f g iv e n by th e p a tc h in g P ic

w ith then

• t^ .

(X)

i s a c o n tr a v a r ia n t fu n c to r w ith r e s p e c t t o X. G iven f* th e re i s a homomorphism o^. ----- o * , hence an induced

►Y ,

homomorphism o f H1 , s. then

i s a commutative group—t h is i s

More d i r e c t l y ,

t h e i r p rod u ct i s resp ect to

.

f* ( £ ) =

More

0 ^ £

g iv e n by the c o - c y c le

is s^ j

d ir e c t ly , i f

£

an i n v e r t i b l e

sh ea f on

X;

(U ^ ),

then

w it h r e s p e c t to

b y th e c o - c y c le

w ith r e s p e c t t o

i s an i n v e r t i b l e

( f “ 1 ( U ^ )).

sh ea f on

and i f

£

f* ( £ )

Y,

is i s g iv e n

N ote a ls o th a t s e c ­

tio n s s e r (Y ,

£ )

induce s e c tio n s f* (s ) C)

Suppose

Then a lth o u g h

s

sense to say of

£x

and

the v a lu e

g (x )

In p a r t ic u la r ,

.

i s a s e c t io n o f an i n v e r t i b l e

does n o t have v a lu e s a t p o in ts

s (x ) £x ,

s

€ r (X , f * ( £ ) )

=0

or

and i f € K (x )

s (x ) ^ 0.

s

0

£ it

on

X.

does make

Namely, i f we choose an isomorphism

correspon ds to

o f gi s

sh ea f

x e X,

g e £x ,

then a t l e a s t whether

o r n o t i s in dependent o f the isom orphism .

one has th e su bset o f

X:

Xg = tx € X | s ( x ) ^ 0) w hich i s e a s i l y seen to be open.

These open s e ts in c lu d e as s p e c ia l cases

30

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

the open s e ts P roj

(R)

used t o d e fin e the to p o lo g y b o th o f Spec (A ) and

( c f . below ( i v ) ) .

R etu rn in g t o P r o j (* )

Rn

is

spanned, as

Then we fin d th a t P r o j i)

X = P roj

[P r o o f: th a t a l l f

in

ii)

( R ) , assume t h a t : RQ-m odule,

If

x e X - U Xf , a re i n

pj

Xf n Xg,

f/ g

is

of

ox on

If

a

p

Rn C

C R such

p , c o n tr a d .]

T h e re fo re th e c o v e r in g

(X f ) (R ),

s h e a f.

0( 1 ).

T h is i s c a lle d o (i)® * 1

of

o (1 ),

one

homomorphism ■r (X , o ( n ) )

o (n )

i s d e fin e d by th e c o - c y c le

k € R ,

then

k

(f/ g )n

g iv e s r i s e t o th e s e c tio n s

d i f f e r p r e c i s e l y by f a c t o r s

th ey p a tch up as s e c tio n s o f

iv )

to

and

the nttl te n s o r power

Xf ; s in c e th ese

X f n Xg,

correspon ds C p,

the g e o m e tric s ig n ific a n c e o f the graded r i n g

[ C o n s tru c tio n : {X f } .

f €R1 .

d e fin e a 1-C zech c o - c y c le on P ro j

Rn

c o v e r in g

Ri

i s a u n it .

f/ g

has a c a n o n ic a l

w hich i s

Xf , f o r

then x

Thus

hence an i n v e r t i b l e I f o (n )

R ^ T T T ^ I r ., .

(R ) has more s tr u c tu r e :

and th e u n its

iii)

by

(R ) i s c o vered b y

R1

On

nx

One checks t h a t , f o r th e to p o lo g y on

X n, % d e fin e d as in cpn (k )

f o r the k / f31

( f / g ) n on

£ (n ) . ]

k € Rn ,

X = P roj

R.

th e open s e ts

(R)

Xk

d e fin in g

a re the same as th e open s e ts

C) above.

L e t us a p p ly t h is new in fo r m a tio n t o study th e s tru c tu re o f the fu n c to r s

hpr o j ^

•

G iven an S -va lu ed p o in t S —

o f P roj

( R ) , one o b ta in s on

P u ttin g t h is f u n c t o r i a l l y ,

S

►P r o f (R )

an induced i n v e r t i b l e

k p ro j (R ) T h is i s i n t e r e s t i n g from two P ic .

P lc

s ta n d p o in ts :

o f th e fu n c to r o f p o in ts o f a P r o j ; the fu n c to r

f* (o p ))

on S.

and i t

•

i t e x p la in s th e n o n - t r i v i a l i t y i s a b e g in n in g in r e p r e s e n tin g

A lth ou gh i t may seem stra n ge to v iew P r o j

as approxim ate group-schem es, r e a l l y r e p r e s e n tin g P ic , cu ra te in th e c a te g o r y (H o t ).

c pn ^

c p„ »

( R ) , or

pn ,

t h is i s q u it e a c ­

Here we have the CW-complex

p r o j e c t i v e n -sp a ce) and hence

sh ea f

one has a v e r y b a s ic morphism o f fu n c to r s :

c Pn

(com plex

P r o j AND INVERTIBLE SHEAVES

31

fu n c to r re p re s e n te d L^y C P* J

fu n c to r r e p r e s e n te d ! Lby C P„ J

l\ \

fu n c to r ~ S - H (S , z ) . l\\

fu n c to r S group o f t o p o lo g ic a l e q u iv . c la s s e s o f l i n e bundles on S v ia

c P^ s E ile n b e rg -M a c la n e Space

K( Z , 2)

.

We can now g iv e the e x p l i c i t d e s c r ip t io n o f the fu n c to r we have been d r iv in g a t .

X± € correspon d as in Then f o r a l l

rn

to

S

p.

i n th e

R1-component o f

£

=

f* (o (1 ))

s±

=

f * (X ± ) e r (S , £ )

z [X Q, . . . ,

Xn 3.

T h is g iv e s an isom orphism :

(S )

/ (£

> s n>

■'

£ an i n v e r t i b l e

sn)

sh ea f on S

sQ, • • •, sn s e c tio n s o f th e re i s an S± (x )

P r o o f:

Not a d i f f i c u l t e x e r c is e ,

by a c o l l e c t i o n

g e th e r ;

which

n' 0( 1 ))

r(

such th a t f o r a l l

g iv e n

•n

one o b ta in s :

P r o p o s it io n 3:

K

(iii)

h

Let

s in ce

( PR) X>

i

£

x e S,

modulo

such th a t

isom or phism.

f o

( c f . EGA 2, §1+) ;

f : S -*•

f . : SQ— ► ( pn) Y , 0 < i < n, w hich p a tch jL i ” — i s a f f i n e , use Theorem 1, L e c tu re 3.

Pn

is

to -

A n ic e C o r o lla r y t i e s t h is in w ith the elem en ta ry d e f i n i t i o n o f P r o j.

space o v e r a f i e l d

a lo c a l r in g C o r o lla r y ;

k—e x c e p t we may as w e l l a t le a s t r e p la c e

k

by

Pn

is

_o: If

£

i s a lo c a l r in g ,

the s e t o f o -v a lu e d p o in ts o f

isom orp h ic t o : { ( aQ, . . . ,

e o , n o t a l l a ^ i n th e max. i d e a l

ofn) |

. P r o o f:

S in ce Spec (£ )

i t s e l f is

c o n ta in in g the one c lo s e d p o in t , i t v e r tib le

s h e a f,

.°3 pec

•

a l l u n its x. e a*

the o n ly open subset o f Spec ( o )

f o llo w s th a t Spec (£ ) has o n ly one i n ­

Sin ce the automorphisms o f

c i s e l y m u lt ip lic a t io n s by u n its o f P r o p o s it io n 3.

),

m)

Ogpec

Pr e

x. e £ * , the C o r o lla r y i s a s p ecT a l case

32

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE As a f i n a l p o in t , i t

i s i n t e r e s t i n g t o g iv e th e g e n e r a liz a t io n o f

t h is l a s t P r o p o s it io n t o Grassmannians.

B e fo re d e fin in g the a c tu a l G rass-

mannian e x p l i c i t l y , we can c h a r a c te r iz e i t by g iv in g i t s fu n c to r : D e fin itio n :

A sh ea f

g

of

o^-modules i s

th e re e x i s t s an open c o v e r in g

X

l o c a l l y f r e e o f rank

{UjJ

of

g lui

S ^ lu ± •

r

if

such th a t

Then the fu n c to r i s : r l o c a l l y f r e e sheaves

Q

■
• • * > r-1

f o r any sequences THEOREM :

i 1, . . . ,

1 ® Pi 1 j i 2>

i p_ 1

and

j^ ...,

1 . 1 > >Jr+ i j p+1

=

0

.

The above morphism from the Grassmannian fu n c to r t o the

fu n c to r o f p r o j e c t i v e

space i s i n j e c t i v e and i t s image c o n s is ts p r e c i s e l y

o f the S -va lu ed p o in ts o f p r o j e c t i v e space s a t i s f y i n g ( # ) . P r o o f:

An S -va lu ed p o in t o f the Grassmannian can be reg a rd ed as

a s u r je c t iv e homomorphism: „n+1

Og

CP

cr . r\

---- * -► g -*• 0

Up t o isom orphism , t h is p o in t i s determ ined b y th e k e r n e l o f the k e r n e l i s a subsheaf o f a f i x e d d eterm in ed g l o b a l l y . T h e re fo re

s h e a f,

cp;

i f i t i s g iv e n l o c a l l y ,

the r e s u lt f o llo w s i f , g iv e n

sin c e i t is

any S -va lu ed

Proj AMD INVERTIBLE SHEAVES

33

p o in t of p ro je c tiv e space s a tis fy in g (# ), th e re i s an open covering of S such th a t over each open su b set, the S-valued p o in t l i f t s u niquely to a p o in t of the Grassm annian. T h erefo re, we can p ass to an open s e t where a fix ed Plticker co o rd in ate i . e . , th is p g en erates £ g lo b a lly . The re la tio n s (#) can then be "so lv ed ," and one checks th a t they take p re c is e ly the form F (• • • *P-^ j_ j * * *) • • • >Jr ^ , . . . , 1 ^ )«-! 1 1 , . . . ,-L k , . . . ,J -r , J ,

where a t le a s t two of the j ' s are n o t in the s e t , . . ., i and where P i s a homogeneous polynom ial of degree N in the r(n+1 - r) fre e v a ria b le s p. ? . On the o th er hand, fo r the S-valued p o in t cp of the Grassmannian fu n c to r to induce a p ro je c tiv e p o in t where p. . ^ 0, i t i s n ecessary and s u ffic ie n t th a t s. = cp(e. ) , . . . , s.

= cp(e. ) i s a b a s is of the sheaf g . Then the id e a l which i s the r r k ern el of © has a uniaue b a s is of the form: k=1 * (where eQ, . . . , en i s the stand ard b a s is of o^+1) . In term s of a^k, the Plucker co o rd in ates come o ut: Pi ± -j a jk, t = (-1) r - k -L 1 9 * * * 9 -L ] £ > * * * 9 J 9 ^- 2 9 • * • j i p

T herefore th e re i s one and only one choice of a^k e r(S , og) correspond­ in g to the given Plucker co o rd in a tes. QkIj C o ro llary 1 : The Grassmannian fu n cto r i s re p resen ted by t i M re la tio n s ) , G„ „ = P roj z [ . . . , p.^ 9 - - - 9 ±4r , • • • ] / (Q uadratic # C o ro llary 2 : The open s e t of G^ „ where p. . ^ o i s isom orphic r to a ffin e space of dim ension r(n+ l - r ) .

APPENDIX TO LECTURE 5 A fu r t h e r developm ent o f the th e o ry r e v e a ls th a t the o p e r a tio n P r o j , as d e fin e d ab ove, i s z a t io n l e t

R =

Rn

S -a lg e b r a ; as such i t

o ft e n to o s p e c ia l.

be a graded r in g .

To understand the g e n e r a li­

Suppose

RQ

happens t o be an

g iv e s a q u a s i-c o h e re n t sh ea f 00

R = 1 n=o

of

^ -m o d u le s on

graded sh e a f o f is

«n

X = Spec ( S ) . o^-a lg e b r a s

Here

R

i s a c t u a lly a q u a s i-c o h e re n t

(a m ou th fu l, but sim ple en ou gh ).

The p o in t

th a t one can en cou n ter such sheaves even on n o n - a ffin e schemes

Thus say

R =

a f f i n e open

Rn

is

such a c re a tu re on some scheme

U C X,

X.

X.

Then f o r a l l

^ r (U ,

R)

=

£

r(TJ,

Rn)

n=o i s a graded r i n g o v e r R)

r (U , £x ) .

T h e r e fo r e one g e ts a scheme

P r o j[r (U ,

to g e th e r w it h a morphism 7T : P r o j r (U ,

R ) — ►U .

One checks ( c f . EGA, 2, §3) th a t th ese p a tc h to g e th e r c a n o n ic a lly t o a scheme

( R) t o g e th e r w ith a morphism: 7r : Proj^ ( R) —

The f o l l o w i n g i s s h ea f o f ran k power o f

E

r (a s

X .

th e most im p orta n t exam ple: on a scheme

X.

Put

£x -m o d u les), and

P

T h i s scheme g e n e r a l!z e s

Pn

Pn

^

Rn

Let

R = E Rf l .

=

i t s e If:

P[ ^ 35

E

be a l o c a l l y f r e e

eq u a l t o the

^

n^*1 symmetric

Then one w r it e s :

*

i.e .,

Xi ' ^Speo z ]

•

36

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

On the o th e r hand, i t i s n ot much more co m p lica ted than i s isom orph ic t o the f r e e then o ver

sh ea f

on

Pn,

°P en c o v e r in g

fo r i f (U^)

X,

U^: P (E )| n

s

P ((o x ) r )ly

s

p r _i

x

[T h is fo llo w s from the g e n e r a l f a c t th a t i f

f :

and

o ^ -a lg e b r a s , then:

R

E of

i s a qu asi-coh eren t, graded sh ea f o f Proj^ ( f * ( R ) ) s

X -► Y

i s any morphism,

x X .

c f . EGA 2. § 3 . 5 . ] F or

p (E ),

0( 1 )

i s co n s tru c te d e x a c t ly as b e fo r e , and one fin d s a

c a n o n ic a l homomorphism: E (if

tr

is

the p r o je c t io n from

induced homomorphism on

P (E )

s u r je c t iv e .

lift in g

onto the base

X ).

M oreover, the

P (E ): tt* (E )

is

ir * (o (1 ))

- ► o (l)

Now suppose a morphism

g : S -► X

i s g iv e n .

Then t o any

h: P (E )

S ------------- ------------ X we can a s s o c ia te th e i n v e r t i b l e

s h ea f

L = h * ( o ( 1 ) ) , and a s u r je c t iv e homo­

morphism: cp:

g * (E )

= h*(7r*E) - * h * ( o ( 1 ) )

An easy g e n e r a liz a t io n o f the r e s u l t f o r

Pn

= L .

s ta te s th a t t h is s e ts up a

f u n c t o r ia l isomorphism betw een the s e t o f S -va lu ed p o in ts liftin g

g , and the s e t o f

L

and

cp.

h

of

P (E )

LECTURE 6

PROPERTIES OP MORPHISMS AND SHEAVES s

1

A f f i n e co n c e p ts :

R-m odules,

Let

X = Spec ( R ) .

M, one can d e fin e a sh ea f M

r(Xf, M) = M( f ) ,

We r e c a l l

of

a ll

th a t f o r a l l

o^-m odules, v i a : f € R .

T h is d e fin e s a f u l l y f a i t h f u l and e x a c t fu n c to r : C a teg o ry o f ] R-modules J [i.e .,

Homo

(M, N) s

C a tego ry o f sheaves o f o^-modules

Hom^(M, N ) ,

and

is exact i f

-X 0 —►M—► N - ^ P —►O

A sh e a f

D e fin itio n : m orphic t o Exam ple:

is e x a c t].

y of

M, f o r some Let

R

o^-modules i s

R-module

y

q u a s i-c o h e re n t i f

is is o ­

M.

be a d is c r e t e , rank 1 v a lu a t io n r in g w ith q u o tie n t f i e l d

K. Then th e re a re two nonempty open s e ts i n S p e c (R ): the whole space X, and the g e n e r ic p o in t i t s e l f U. A sh e a f y o f

o^-modules c o n s is t s , t h e r e fo r e ,

in a)

an R-module

b)

a homomorphism o v e r

A = y ( X ) ; a K -v e c to r A

T h is

y

i s q u a s i-c o h e re n t i f

space

B = ^ (U ),

R B .

and o n ly i f : B = A K . R

THEOREM 1 :

If

X

i s a f f i n e , and

(A )

y

(B)

H ^X, 5 ) = (0 ),

is

spanned, as

if

Wecan now g e n e r a liz e th ese D e fin itio n :

Let

c o h e re n t, i f

e q u iv a le n t ly :

i)

X

be a

y |TT i

i s q u a s i-c o h e r e n t, then s e c tio n s

r (X , 5 ) ,

i > 0. con cep ts in v a r io u s

scheme. A sh e a f

th e re e x i s t s a c o v e r in g such th a t

y

o^-m odule, by i t s

(U^}

yo f of

i s qua s i- c o h e r e n t ; 37

ways:

o-^-modules i s q u a s iX

by a f f i n e

open s e t s ,

38

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE ii)

YU C X,

U a f f i n e and open,

i s q u a s i-c o h e re n t.

A v e r y u s e fu l a p p lic a t io n o f t h is con cept i s in : P r o p o s itio n - D e f i n i t l o n :

Let

i s a l o c a l r in g e d space

Y

subspace o f

X

be a scheme.

A c lo s e d sub-scheme

X, and whose sh ea f o f r in g s

one has

(0

e q u iv a le n t ly

0

is

i s a q u o tie n t o f

a sh ea f o f id e a ls in

q u a s i-c o h e re n t, o r

The f a c t th a t i f

Y C X

whose u n d e rly in g t o p o lo g ic a l space i s a c lo s e d

Y

Y

is it s e lf

i s a scheme, then

0

o^:

i.e .,

o ^ ) , p ro v id e d th a t a scheme.

i s q u a s i-c o h e re n t comes

from : P r o p o s it io n 2 ; if

U C Y

in g ) .

is

Let

►Y

be a qu asi-com pact morphism o f schemes ( i . e .

open and a f f i n e ,

Then i f

R ^ f* (y )

f

X

y

f ” 1 (U)

adm its a f i n i t e a f f i n e

i s a q u a s i-c o h e re n t s h ea f on

a re q u a s i-c o h e re n t on

a re th e schemes

open c o v e r ­

a l l the sheaves

Y.

One fin d s , from the above d e f i n i t i o n : X = Spec (R )

X,

the c lo s e d subschemes o f

Y = Spec ( R / l ) ,

f o r id e a ls

ICR.

We

a ls o make th e d e f i n i t i o n : D e fin itio n : of

X,

f Y ---►X

If

then

D e fin itio n :

f

is

Let

X

i s an isom orphism o f

be a scheme.

scheme o f an open subset of

Y

One o f the

( nX red u ced ” ) . id e a ls

0

is

A sub-scheme

U C X.

w ith a subscheme o f

Example:

Y w ith a c lo s e d sub scheme

a c lo s e d im m ersion. Y fC X

An im m ersion

Y

i s a c lo s e d sub-

►X

i s an isomorphism

X.

most im p orta n t sub schemes o f a scheme X i s

As a c lo s e d su b set,

XL^^ = X,

but i t s d e fin in g sh ea f

of

the subsheaf: r (U , 0) = (s € r (U , £x )| E q u iv a le n t ly , sv € o x -x

One checks th a t i f I

U = Spec ( R ) ,

then

0

= (a € R | E q u iv a le n t ly , a or

a

s (x )

= o, a l l

x e U

i s n ilp o t e n t , a l l x € U) is €

the sh ea f

I

, where

e v e r y prim e i d e a l

is n ilp o te n t). o

T h e r e fo r e ,

0

i s q u a s i-c o h e re n t.

(Compare L e c tu re 3,

1 ).

Another g e n e r a liz a t io n o f the con cept o f ’’a f f i n e " i s : D e fin itio n : i)

A morphism

f

►Y

is a ffin e i f

e q u iv a le n t ly :

th e re e x i s t s an a f f i n e open c o v e r in g f " ' 1^ )

ii)

X

V

is a ffin e ,

fo r a l l

a f f i n e open s e ts

V C Y,

(U^)

of

i; f ” 1(V )

is a ffin e .

Y

such th a t

PROPERTIES OF MORPHISMS AND SHEAVES C o r o lla r y o f Theorem 1 :

If

modules i s q u a s i-c o h e r e n t, (A )

f X ---- ►Y

39

i s a f f i n e , and the sh ea f

5

of

o^-

then:

the c a n o n ic a l homomorphism: f * (f * S) is

?

s u r je c tiv e ;

(B )

= (o ),

fo r

i > o.

The con cep ts o f f i b r e p rod u ct and a f f i n e morphisms a re connected by the v e r y sim ple but im p o rta n t: P r o p o s it io n 3 :

f X ----- ►Y

Let

be an a r b it r a r y morphism.

be an a f f i n e morphism, and l e t

We w r it e

X1

fo r

X x Y’ y

l a b e l l e d as f o llo w s :

Y1

g

►Y

w ith morphisms

g Then

f 1

i s an a f f i n e morphism.

And i f

F

i s a q u a s i-c o h e re n t s h ea f on

X, )

a ( c a n o n ic a lly )

U g '* (

j

)

•

o

2 fin ite

We d e fin e s e v e r a l con cep ts by s p e c i a l i z i n g the above t o a more

s it u a t io n :

D e fin itio n :

A scheme

i) ii)

i s n o e th e ria n i f ,

e q u iv a le n t ly :

open a f f i n e c o v e r in g

such th a t

i s n o e th e ria n ;

X

r(U ^ , o^)

{U^}

i s qu asi-com p act, and f o r a l l open a f f i n e

r (U , o^) iii)

X

th e re e x i s t s a f i n i t e

of

X

U C X,

i s n o e th e ria n ;

the ord ered s e t o f c lo s e d subschemes o f

X

s a t i s f i e s the

d escen d in g ch ain c o n d itio n . D e fin it io n :

A q u a s i-c o h e re n t s h ea f

co h eren t i f ,

e q u iv a le n t ly :

i)

th e re e x i s t s an a f f i n e rtU ^ ,

ii) N o te .

Q u a si-coh eren t —A

x

is

(0 ),

D e fin itio n : fin ite

if

is a

is then

open c o v e r in g

{U^}

open

of

X

X

is

such th a t

ty p e ;

U C X .

subsheaves and q u o tie n t sheaves o f coh eren t sheaves c o h e re n t; i f

the s t a lk

5

o f a coh eren t sh ea f

X

$ ^ (0 ) in a n eighborhood o f

An a f f i n e morphism

e q u iv a le n t ly :

on a n o e th e ria n scheme

rCCJ^, o^) -module o f f i n i t e

same f o r a l l a f f i n e

a re c o h e re n t; , ov at

5 )

y

X

f

►Y ,

where

Y

x. i s n o e th e ria n , i s

SF

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

14-0 i)

f* (£ x )

ii)

f

i s coh eren t on

is

o f fin ite

co h eren t P r o p o s it io n p o in ts

If

f ” 1(y )

y X

on f

Y;

typ e (hence X,

►Y

f* ( 5 )

is fin it e ,

is fin it e ,

X

i s n o e th e ria n ) and f o r a l l

is

coh eren t on

then f o r a l l

Y.

y € Y,

the s e t o f

( t h i s p r o p e r ty i s what G roth en d ieck c a l l s

" q u a s i- fin ite ” ) . P r o o f:

If

A = f* (£ v )v

s c h e m e -th e o re tic f i b r e co h e re n t,

A

H (y ),

y

-y

f “ 1(y )

is

then i t i s e a s i l y seen th a t the

sim p ly Spec ( A ) .

i s a f i n i t e d im en sion a l

But s in ce

K (y ) - a l g e b r a , hence

f* (o x ) is

Spec (A ) i s

fin ite . QED C oncerning th e to p o lo g y o f n o e th e ria n schemes, the key p o in t i s th a t th ese a re n o e th e ria n t o p o lo g ic a l spaces, f o r c lo s e d su b sets.

i.e .,

s a t i s f y the d . c . c .

C on sequ en tly, e v e r y c lo s e d subset i s a f i n i t e union

o f ir r e d u c ib le c lo s e d su bsets which a re c a lle d i t s

components.

T h is i s ,

o f co u rse, the g lo b a l t o p o lo g ic a l a n a lo g o f th e d ecom p osition o f an i d e a l i n a n o e th e ria n r i n g in t o an in t e r s e c t io n o f p rim ary id e a ls .

The f i n e r

a s p e c ts o f the d ecom p osition theorem come in v i a the o p e r a tio n " A " : D e fin itio n :

Let

?

be a coh eren t sh ea f on a n o e th e ria n scheme

A ( 5 ) = (x € X| 3

a s e c tio n an i d e a l

s € J x I C ox

id e a l, i . e . , and is

th a t

which i s a n n ih ila te d by p rim ary t o th e maximal

an open neighborhood

of

x,

s

x)

f o r a thorough d is c u s s io n o f t h is con­

I t fo llo w s im m ed ia tely from the d ecom p osition theorem f o r modules A( ? )

is a fin it e

s e t.

M oreover,

A( J )

in c lu d e s in p a r t ic u la r ,

the g e n e r ic p o in ts o f e v e r y component o f the support o f subset o f

X )- b u t, in g e n e r a l, i t

p o in t s . " Z

U

such th a t the support o f

the c lo s u re o f

[ c f . BOURBAKI, A lg . Comm., Ch. k, c e p t ].

3

s € r (U )

X.

On the o th e r hand, i f

y

(a s a c lo s e d

a ls o in c lu d e s ’’embedded a s s o c ia te d Z

i s a c lo s e d subset o f

X

and we make

in t o a c lo s e d subscheme v i a the sh ea f o f a l l fu n c tio n s which a re e v e r y ­

where

0 on

then A (o z )

Z

(t h is

i s known as the reduced subscheme s tru c tu re

i s p r e c is e ly

on

Z ),

the s e t o f g e n e r ic p o in ts o f the components o f

Z. o

3

F la tn e s s :

D e fin itio n : sh ea f o f

s io n

n
.

—X

fp

.

x* e X

W ith r e s p e c t

k3

PROPERTIES OF MORPHISMS AND SHEAVES

f*(v) € % C y ,

s in c e e x a c t ly th a t

in .

th e in v e r s e image o f the maximal i d e a l

f ( x ! ) = y.

[i.e .,

c-*

i

Spec (o y )

x

i

Y

]. x 1 € A( J ) .

The p r o p o s it io n w i l l t h e r e fo r e be proven i f we v e r i f y th a t ^£x i

is

use the diagram :

Spec ( o )

But

mx ,

By the rem ark f o l l o w i n g Theorem 1, L e c tu re 3, t h is means

i s p rim a ry f o r th e maximal i d e a l

the induced s e c t io n

s’ €

m^., C £x i ,

and i t

k ills

?x i • QED

Example 3 :

Now c o n s id e r

th e case o f a f i n i t e morphism

f

e t h e r ia n , and a c o h eren t s h e a f

f

X

on

X.

The c o n t in u it y

?

is

l o c a l l y f r e e on

-► Y ,

of

Y no­

?o v e r

Y

e x p re s s e s i t s e l f as f o l l o w s : P r o p o s it io n 7: [ ? fla t P r o o f: X = Spec ( A ) , Let M

? is

over

[f*

The r e s u l t b e in g l o c a l on where

A

Y,

/B,

M.

hence f o r a l l prim e id e a ls

i.e .,

suppose

i s a B -a lg e b r a , and i s

corresp on d t o the f i n i t e A-module fla t B^,

fin ite

/Y] < = >

f* ( 5 )y =

If

typ e o v e r a n o e th e ria n

?

oy = Bp

.

Y = Spec ( B ) ;

o f fin ite

fp C B,

i s f l a t over

Y]

typ e as

is fla t

/Y,

Mp = M .

then

B-module. then is f la t

But a module o f

l o c a l r i n g i s f l a t o n ly i f i t

is

fr e e .

T h e r e fo r e , th e re i s an isomorphism

4 of

oy -m odules.

og in some n eighborhood o f at

y,

~

r*r

But such a homomorphism i s induced by a homomorphism:

y;

—*

f*(f)

and th e k e r n e l and c o k e rn e l, h a vin g

a ls o v a n is h i n a n eighborhood o f

y.

T h e re fo re

f* ( 5 )

0 s ta lk s is lo c a l­

ly fr e e . The co n verse i s z a t io n o f th e

c l e a r , s in c e the s t a lk

£ f ( x ) -module

y

at

x e Xi s a l o c a l i ­

f # ( JF ) f ( x ) • QED

Example k : Y y,

We s h a ll fu r t h e r a n a ly ze th e s it u a t io n o f Example 3, i n case

i s reduced and ir r e d u c ib le .

Suppose

one has the diagram :

y e Y.

V ia th e f i b r e

„

i

| Y

i

Spec

K (y )

of

f

over

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

and 5 on X induces a sheaf J on Xy. Algebraically, if Y = Spec B, X = Spec (A), and 5 corresponds to the A-moduleM, then y comes from a primeideal p C B, K(y) is the quotient field of B / p , =

X

* 7

Spec (A 0 B

=

KCy))

y

A

Since A isafinite B-module, mutative algebra over K(y).

H(y))

B

A ^ H(y)

= M^0 B

is a finite

H(y) • dimensional com­

Note first of all that (*)

r(x , 7 ) J

J

s f #(5) 0 K(y) s * M 0 Oy B

K(y) .

(Cf. Proposition 3 of this lecture.) Proposition 8 : [ ? flat /y] < = >

[the function

y

dim j((y)f*( y ) 0

K(y)

is constant] . Proof: The " = > " follows from Proposition 7 , Y being irreducible and hence connected. To prove it suffices to show that for all y € Y, f*(? )y i s a free o^-module. Lemma: Let A quotient field

be a noetherian local domain with residue field K. Let M be a finite A-module. Then

k,

and

[dinv M 0 K = dim, M 0 k] = > [M a free A-module] . K A k A Proof: Note that if m C A is the maximal ideal, M k ^ M/m • M. Let f 1,..., fn be elements of M whose images f^ in M/m • M form a basis over k. Then the f^ define a homomorphism cp: 0 — L — An — 2-»M — N — 0

(*) (L and obtain:

N

being the kernel and cokernel resp.). Tensoring with _ k11 M/m • M — N/m • N — 0 .

But cp is surjective since the f^ By Nakayama's lemma, N = (0). Now /A, we obtain:

k,

we

span M/m • N; therefore, N = m • N. tensor (*) with K. Since K is flat

0 — L 0 K — Kn - * M 0 K —►() . A A By hypothesis, K11 and M 0^ K are both K-vector spaces of dimension n. Therefore, L0, K = (0), i.e., L is a torsion module. But since L C n A , this implies that L = (0). QFD

h5

PROPERTIES OF MORPHISMS AND SHEAVES

Example 5 : As a final point, let us consider two completely concrete cases: (I) Y = Spec k[yj X = Spec k[x] 2 y = x . (k alg. closed). Then if

C k[y]

is the maximal ideal

(y - a ),

k[x]/p • k[x]

s

k[x]/(x - a) © k[x]/(x + a),

a ^ o

k [x ]/p • ktx]

s

k[x ]/(x ) ,

a =

o

and both are commutative algebras of dimension 2 over k. This being a constant f is flat. (One should also check non-closed points of Y.) (II)

Y = Spec k[x2, x.,x2, x2] X = Spec k[x1, x2]

Then if & C k[x.j, x ^ 2, x2 - p2), one finds 2

2 Xg]

is the maximal ideal

2

2

(x1 - oc ,

x i x2

“

k [ x 1, Xg]/«p * k [ x v x 2] s k [ x 1, X g ] / ( x 1 - a, x 2 - p) 0 k [ x 1, X g ] / ( x 1 + a, Xg + p) if

a

or

p / 0,

k [ x 1, Xg] /*> * k [ x 1, Xg] ^ k + x 1 ’ k + Xg-k (x2

= x1 Xg = x2= o)

if

a =

p = o.

The former is a commutative algebra of dimension 2 ; the latter is one of dimension 3. Therefore f is not flat.

LECTURE 7 RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON Pn As above, let P R = Proj Z[XQ,..., X ^ , let o (1) be the canoni­ cal sheaf on Pn , and identify XQ,..., Xn with sections of 0(1 ). For all schemes S, on Pn x S, put £ ( 1) = P*( o (1))

Xi If 5

=

(by abuse of language)

the induced section p*(Xi) (by abuse of language).

is a coherent sheaf on ? (m) =

Pn x S, ?

®

put (o(1

f m) .

£pn * s 1 ° Serre's results. We look first at the readily visualized case S = Spec (k), k a field. Fix 5 (again coherent), and write Pn k for Pn x Spec (k): (i) (ii)

H^ and For H*^

Pn k, 5 ) is finite dimensional over k, for all is*(0) if i > n; all 5, there exists mQ such that if m > mQ Pn k, y(m)) * (0), i > o and 5 (m) spanned, as

i;

— pn ^.-module, by its global sections; (-1) 1 dimk Hi( Pn k, ?(m)) the Hilbert polynomial of 5 . Consider the functor:

(iii) (iv)

00

or :

5 -

©

r( P

m=o Here

5

’

is an object in the category

is a polynomial in

v, e

[

t € r ( Pn k, of

5 (m )),

then

^(m) £(1) s

kl

X^^ • t £(m+1)

5(m)) .

of coherent sheaves on C1

pn k5 and ^ ) is an object in the category XjJ-modules of finite type. If

m -

of graded

i s the section .

k[XQ,...,

t Xi

1+8

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

Take morphisms in to be: Home , (M, N) = liji o pre serving

©

m>m0

1VL

©

'm >mQ

N

1

m j

Then a is an equivalence of categories, especially a is exact, and takes Horn's into Horn's. The key step in proving this is the explicit con­ struction of the inverse of a. This functor is a graded generalization of the ~ operation in the affine case. Start with a graded module M, of finite type over k[XQ,..., X ^ . For each i, form the tensor product. M (1) = M

® k[XQ,..., ^ k[X] 0

J- ] , i

and let be the sub-module of degree o. Then is a module of finite type over the affine coordinate ring kl^/X^,..., Xn/Xi ] of ( Pn)x • One verifies that the sheaves on the affine spaces patch together in a natural way: verse of oc.

the result is called

M

and this is the in­

(v) Before proceeding to generalizations, we want to make some attempt to describe the "yoga" cohomology. The cohomology of sheaves, in a general geometric setting, is just a piece of machinery designed to analyze the connection between the local and global structure of space; viz. given any local data, the set of all such local data will form a sheaf and its cohomology groups are a sequence of invariants describing how "twisted" these data can be from a global point of view. The essen­ tial point is that (a) these groups are almost always very computable, (b) the obstructions to making global constructions are elements of such cohomology groups. In the case of algebraic geometry, the objects of global geometric interest are the global sections of coherent sheaves. These arise for ex­ ample out of the desire to determine how many functions exist on some scheme with prescribed poles; in what projective spaces can a given scheme be embedded; how many global differential forms of a given type exist on some scheme; and in the infinitesimal linear form of many non-linear existence problems. But to compute the vector space of sections of a coherent sheaf y on Pn , the essential difficulty is that r is not a right exact functor. This was realized by the Italian geometers, who worked indirectly but still (as we now realize) very closely with the higher cohomology groups. It should be pointed out that the fancy definitions given cohomology recently—via standard resolutions, derived functors, especially in the category of all sheaves-^which look very uncomputable—are just technical devices to simplify somebody's general theory. One may as well treat the cohomology of a coherent sheaf on Pn just as the satellites of r in

RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON

Pn

the workable category of coherent sheaves. [In technical terms, coho­ mology is effacable in this small category]: e.g., the group H1 ( p1, o p (-2 )) s k is nothing but the cokernel of the sequence: o —► r(

p 1#

o

(-2)) -► r(

p 1#

o p^-1))

r( Pl, H(x))

coming from the exact sequence of sheaves: 0 -

o

“

(-2 )

0 X.

----- — ► o p

*1

(- 1 ) -

K (x )

-

o

rl

on P1, where H(x) is the sheaf with support only at the point x whereX 1 = o, given by the module which is the residue class field of —x*

We must recall, for future use, the facts about the cohomology of £ p (m) itself: n H3^ Pnjk,o p (m)) = (o), if o < i < n (0), = (0), =

if i = n,

m > -n - 1

if i = 0 ,

m o

2 ° Grothendieck's globalization. Now suppose S is any noether­ ian scheme, and J is again coherent on pn x S. Let p: Pn x S -► S be the projection. Then: (i) RiP*( T) is coherent for all i; and is (o) if i > n . (ii)

For all

5, there exists

mQ

such that if

m > mQ,

H'Sp-kC ^(m )) = (0), i > 0 , and p p* ?(m) -► ^(m) is surjective* (iii)

Consider the functor: 00

a:

Here

?

? -►

© p*( ^(m)) . m=o is an object in the category £ of coherent sheaves of

qn modules; and a( y ) is an object in the category C 1 of quasi-coherent sheaves of graded £g[XQ, X-j,..., XR ]-modules of finite type-where the morphisms are given by: Horn e , ( * , * ) =

HomG ra d a tion 0

Then

a

m> m0

Jlks

©

m >m0

i s an e q u iv a le n c e o f c a t e g o r ie s . In f a c t ,

the in v e r s e ~ t o

s t a r t w ith the s h e a f say

©

p r e s e r v in g

£

S = Spec (R ) .

on Then

)R

S.

a

is

c o n s tru c te d e x a c t ly as in 1 ° :

F or s i m p lic it y , assume

i s n o th in g but a graded

S

is a ffin e ,

R [X Q, . . . ,

-module

50

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

of finite type.

For all i,

put

^7 ])

= degree o component of ( % R[XQ,..., R [X] Then

i

)R is patched together out of the sheaves

on:

r Xo Xn 1 Spec R|_jq;’- " ’ X ^ J = ( p n >< s)x^

.

3° Connection of higher direct images with cohomology on the fibres. The principle difficulty in using the results of 2 ° is in re­ lating R^p*( 5 ) to the cohomology along the fibres of p. Thus, if s € S, let pn s = t h e fibre of p over s, and let ? induce the co­ herent sheaf ya on p„ Is there any connection between; S Llj 3. rS

*^)

and

® K(S)

H5^ Pn s , fg) .

This is a special case of the more general problem; sion" g: T S, look at the diagram: Pn x T----- —

given a "base exten­

. PR x S

What is the relation between S*R1 P*( 5 ) and R^-q^h* ? ) , for coherent sheaves has homomorphisms:

on

5

Pn x S?

But, for any open set

U C S,

one

pn x U, f ) - H1( Pn x g' 1 (U), h* J ) - H°(g_1 (U), R1 q*(h* 5) ) hence a homomorphism:

hence a homomorphism:

*

1

1

*

g R p*( ? ) -► R q*(h 5 ) .

If, for every g, this thi is an isomorphism, we shall say that commutes with base extension. First of all, there is a simple "stable" result when twisted sufficiently: (i)

For any 5, and any if m > mQ, then:

a

T —^

S,

there is an

J

R^p*

has been

mQ such that

g*p#( y(m))-^~ q*h*( ?(m)) (of course, both sets of higher direct images are zero).

RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON

PR

51

Idea of proof: This really asserts nothing more than the compati­ bility of the equivalences of categories and with tensor pro­ ducts. Thus, over _S, y is defined by thesheaf of graded Og[X0,..., -modules: ^ «c( ? )= ft = © P*(-y(m)) D

and, over modules:

T,

h* y

m=0

is defined by the sheaf of graded a„(h*

5 )=

ft

=

1

e q*[h*( m=o

o,p[X0,..., ^ 3 -

5 (m)) ] .

One wants to know that the natural homomorphism from g* 5R to ft is an isomorphism in our funny category (where any finite number of graded pieces canbe ignored). To prove this, use the inverse ~ to a I Since and «T are equivalences of categories, it suffices to prove that g* % * h*( w )

.

But this is an immediate consequence of the definition of ~ [for de­ tails, cf EGA, Ch. 2 , §§2 .8 . 10 when S, T affine; 3.5.3 in general]. However, to obtain really precise relations between these higher direct images, we must look at the case when y is flat over S; (ii) Assume y is flat over S, and that for some i, and some sQ e S, the homomorphism: r

S . C ?

) ®

H(S

u

) -

p

IA>ao

,

y

0

)

is surjective. Then there is an open neighborhood U of sQ in S such that for any base extension g: T -*■ U, the homomorphism g V p ^ y ) is an isomorphism. (iii)

R^t^y)

(See EGA, Ch. 3, §7.7.)

With the same assumptions as in homomorphi sm: R1 _1 p*OF) ®

(ii), it follows that the

K(s0) - H 1- ^ Pn > v 5 Sq)

is alsosurjective if and only if Rip*( y) is a free sheaf of o^-modules in some neighborhood of sQ . (See EGA, Ch. 3, § 7.8.) Corollary 1 : In the flat case, if H^+1( P.. , y ) = (0), then there ---------n, sQ sQ is an open U C Scontaining sQ such that, for g: T -*■ U: g*R j p * (

Rjq*(h*5 ) .

In p a r t ic u la r :

R jP*( 5) ®

H(s) — * HJ( pn g,

?s) ,

52

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

Proof: C o r o lla r y 1^: then

Use (iii) for

i = j+1

In the f l a t c a se, i f

pn s9

for a 1 1

yB) =

and then

R'Lp * ( ? )

s e S,

=

(ii) for i = j. (0 ),

and all

fo r a l l

i >

i Q,

i > iQ .

Proof: Apply Corollary 1 first for j = n to prove that Hn ( Pn g, 5 ) = (o), all s € S; then for j = n- 1 to prove that B11-1 ( Pn>s, 7 S) = (0), Corollary 2 : homomorphism

all

s e Sj

etc.

In the flat case, givena coherent sheaf g on cp from g to p*( 5) such thatthe induced

s ® K(s) - H°( is an isomorphism for all ly free sheaf, and

s,

then

Pn s ,

and a

Js)

cp is an isomorphism,

g*P* 7 for all

S,

g

is a local­

-n - 1 = free sheaf of o^-modules, with basis given by monomials in XQ,..., Xn of degree m, if i = 0 . Proof:

Use

3°

QED

(ii) and (iii) and 2 ° (v).

it seems worthwhile to give one non-trivial example of this

k°

theory: Let n = 1, S = Spec k[t], k an algebraically closed field P 1 x S = Proj k[ t j XQ, X 1 ]; let R = k[t;XQ-, X 1 3 . Forall integers m, and graded R-modules M, put M(m) equal to the R-module such that

(i)

(ii)

M (m >k = Define the graded module

(iii)

M

• as

l~R© R © R (-1 )/modulo the element (Xn, X., t) L of degree 1 . Put 3 = M. nel in:

Corresponding to its definition as module,

0 -,' - P 1 x S (_1) where

y = (XQ,

* ^ P,x S ® - P 1x S ® - P 1 x S (_1)

X 1, t) [i.e., tens oring with

— P x s(k)

to

£p

x s ^ ^

Since the map

Let 0 e S be the point ? 0 is defined by:

is the coker­ 0

o p x S (k) to

function

t maps

r

gotten by tensoring tJ with

K(x), (x e P^ x S), is never 0 , it follows that sheaf of rank 2 , and it is flat over S. (iv)

"* ?

X.^ maps

— P x S ^ ^ 1^ 181(1 multiplication by the ordinary

y

y

is a locally free

t = 0 . Then the induced sheaf

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

5^

(-1 )

(X0 , X ,, 0) o rp, —

o p (-0

® — o rp 1

—*i

? o -o

and one checks th a t t h is means: ?o - On the o th e r hand, i f

7* 0,

•( a € k ) ,

p

- p.

( + 1)

s e S

is a

(-1 )

k - r a t io n a l p o in t where

(-1 )

(- 1 )

where

makes

cpg

5s (v )

t = a

then th e d iagram :

i s d e fin e d by the 2 x 3

isom orph ic t o

1

0

0

1

p -o ^

P.

m a trix - V ° -X ,/ a

© o D

The c o h o m o lo g ic a lly in t e r e s t i n g p o in t i s : P, ( J ( - O )

= (o )

H ( P, f Q , ? o ( - 1) ) i.e .,

p#

which

is

= k ,

does n o t map onto the

H°

a lo n g th e f i b r e ;

c o n s is te n t w ith the th e o ry i n v ie w

o f:

H'( p1>0, S 0 ( - D ) * k R1P * ( ? (-■ > )) = Uq , i.e .,

th e sh ea f c o n cen tra ted a t

i s the re s id u e c la s s f i e l d

k

t =0, of

w hich as

module

Oq g .

[P ro v e t h is by s e t t in g up an e x a c t sequence

where

Z C P-, x S

3° to compute

is

th e c lo s e d subscheme

R1p# ( 5 ) ,

X1 = o ,

u s in g the r e s u lt s o f

and u s in g th e cohom ology s e q u en ce.]

LECTURE

8

FLATTENING STRATIFICATIONS The problem we want t o c o n s id e r i s ?

on

Pn x S,

th is :

G iven a co h eren t sh ea f g

S a n o e th e ria n scheme—f o r a l l morphisms

T ------- S,

one

has the induced s h e a f: 3g = O p x g ) * ?

on

Pn x T .

Can you d e s c r ib e the s e t o f a l l morphisms T ?

D e fin itio n : S ,,.;., s € S

If

Sm

S

O

such th a t

i s a scheme, a s t r a t i f i c a t i o n o f

o f l o c a l l y c lo s e d subschemes o f

i s i n e x a c t ly one su bset THEOREM:

..., y

g

5 o

To answer t h i s , we f i r s t make:

Sm

of

S

S

S

i s f l a t over

is a fin it e

set

such th a t e v e r y p o in t

S^.

In the above s it u a t io n , th e re i s a s t r a t i f i c a t i o n S ., g T ►S (T n o e th e r ia n ),

such th a t f o r a l l morphisms

i s f l a t over

T

if

and o n ly i f

the morphism

g

fa c to rs :

m T

g'

'

1=1

II

c— S .

We w i l l c a l l t h is a f l a t t e n i n g s t r a t i f i c a t i o n : is

o b v io u s ly u nique.

I f i t e x is ts , i t

There i s an analogous problem when

p la c e d b y any scheme

X

p ro p e r o v e r

S.

Pn x S

is re ­

G ro th en d ieck has then p roven a

s l i g h t l y weaker theorem , but by much d eep er methods. o 1 its e lf. if

it

Look f i r s t a t th e case

n = 0;

is

F ix a p o in t

sfo r

whose im ages in th ese

i s a co h eren t sh ea f on

fF i s sim p ly g * ( 5 ) , and i t i s f l a t o v e r o lo c a lly fr e e o v e r T. F or a l l s e S, le t e(s) = dim K (s )

v i a the

?

Now

a^

a w h ile , ®

le t

e = e (s ),

and

H (s ) a re a b a s is o f t h is

exten d t o s e c tio n s o f a^

K (s ))

( ys

5

55

if

S

and o n ly

.

choose a 1 , . . . ,

a0

v e c t o r space.

Then

in a neighborhood

one d e fin e s a homomorphism:

T

U1

of

s,

€ and

!Fg

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

56

4 - “— in

U1 .

a^

g e n e ra te

S in ce the

ai

in

a (p o s s ib ly ) s m a lle r neighborhood U2

s m a lle r neighborhood its

s e c tio n s o v e r

in

Ug ( f o r

g e n e ra te

5i t s e l f . U g,

Ug,

f) .

4

—

Let

Ug

if e

by Nakayama's

homomorphism of

s.

cp

P a s s in g

Ker (q>)

lemma, the

i s s u r je c t iv e t o an even

i s g e n e ra te d

by

4

y

be c a lle d 5

- o

Us .

i s g e n era ted by

e (s )

s e c tio n s e v e r y ­

hence:

(* ) i.e .,

K (s ),

th e

we may assume th a t

Note f i r s t o f a l l th a t where i n

®

and we have co n s tru c te d an e x a c t sequence:

U g,

some

5

T h e re fo re

*

s ’ € Ua ,

e ( s ')

i s upper sem i-con tin u ou s.

< e (s )

.

T h e re fo re th e s e t

ZQ = i s € S | e ( s ) = e ) is

l o c a l l y c lo s e d .

on ly i f

M oreover, i f

K ( s ') is

s' e U ,

o. T h e r e fo r e ,

t io n s on

Us ,

( ^ i j ) ai i i

j

(* )

T

if

if

^

f

>|r( s 1 )

i s ex p ressed by an

*ias suPPo r t g

►Us

P roof o f * : tio n s

i s e x a c t on

i f and

T,

e;

o f fu n c­

d e fin e d by th e i d e a l Yg

has th e p r o p e r ty :

i s any morphism (T n o e th e r ia n ), then e = e (s )

i f and o n ly i f

0

on

T.

g

Yg

i f and o n ly i f a l l th e fu n c ­

But s in c e the sequence:

t h is i s e q u iv a le n t t o a s s e r t in g th a t

c o n v e r s e ly ,

say

g*(q>) .

g*( y ) is fa c to r s

Yg .

C e r t a in ly t h is in tu rn im p lie s th a t

be the k e r n e l o f

p o in t

= e (s )

e x f m a trix Us

I cla im th a t

g f a c t o r s through

&re

morphism.

of

2e n Ug .

through the c lo s e d subscheme

g

e ( s ')

p

—— — * H ( s ')

th e c lo s e d subscheme

l o c a l l y f r e e o f rank

o f rank

then

th e homomorphism

g* ( 5 )

is

g*(cp)

g * (5 )

is

i s an i s o ­

lo c a lly fre e

l o c a l l y f r e e o f rank e ,

T en s o rin g w ith the re s id u e f i e l d k

and l e t a t any

t € T, one f in d s : T o r 1 ( g* y , k) -*■ g ® k

- ^ k e - * 'g * ( 5 r ) k

0

ii

(o)

Sin ce

g * ( y ) k

i s a k - v e c t o r space o f dim ension

e,

g k = ( o ) ,

FLATTENING STRATIFICATIONS hence by Nakayama's lemma, w here, and

g*(cp)

g = (o )

n ear t .

57

T h e re fo re

g = (o )

every­

i s an isom orphism . QED

Note th a t p r o p e r ty ( * ) a n eighborhood o f any p o in t o f any two p o in ts o f

Z _, e

c h a r a c t e r iz e s th e subscheme

Ze n Ug .

T h e r e fo r e , i f

i n th e open s e t

U n U0 s^ s2

s1

and

Ys s2

in are

the two subschemes

Y

and Y a re e q u a l. In o th e r w ords, the subschemes Y_ p a tch t o 1 *^2 ^ g e th e r t o endow the l o c a l l y c lo s e d subset Z0w ith a s tru c tu re o f subscheme.

C a ll t h is subscheme

t io n o f

S,

Y0 .

The c o l l e c t i o n

and, by v i r t u e o f ( * ) ,

2°

Y_ e

is a s t r a t ific a ­ { YQ)

is

3 °, I want t o make e x p l i c i t th a t we have p roven

more than th a t a f l a t t e n i n g dexed th e subschemes

(Y g )

f o llo w s im m ed ia tely th a t

5.

a fla t te n in g s t r a t ific a t io n fo r For use i n

it

s t r a t i f i c a t i o n (Y e ) e x i s t s :

so th a t

5 %

o^

is

We have even i n ­

l o c a l l y f r e e o f rank

e.

e

B e fo re a t t a c k in g the g e n e r a l case o f the theorem , we

need an e le g a n t p ie c e o f "h a rd 11 a lg e b r a ( c f . EGA, Ch. k f § 6 .9 ) w hich g iv e s us som ething t o s t a r t w ith : f P r o p o s it io n : Let X ►Y

be a morphism o f f i n i t e

?

and ir r e d u c ib le .

Then th e re i s a non-empty open su bset

the r e s t r i c t i o n o f

be a c o h eren t s h ea f on ?

to

P r o o f: set

Spec ( A ) ;

su bsets

Vi ,

th a t a f f i n e le t M.

f

5

B

i s f l a t o ver

U.

i n t o an A - a lg e b r a , and l e t

Y

i s reduced

U C Y

such th a t

U. Y

be some a f f i n e

can be c o v e re d b y a f i n i t e

c l e a r l y s u f f i c e s t o fin d one

open p ie c e

make

X

Assume th a t

i s f l a t over

We may c l e a r l y r e p la c e

and s in ce it

h” 1(U)

X.

typ e o f n o e th e ria n

schemes, and l e t

U

f o r each

Vl

T h e r e fo r e , l e t

5

open sub­

s e t o f a f f i n e open so th a t in

X = Spec (B ),

correspon d t o th e B-module

Then we s h a ll p ro v e :

(* )

th e re i s an elem en t

f

e A

such th a t

= M ^ Af

i s a f r e e Af -

m odule. Note f i r s t th a t i f 0 — L

M— N — 0

i s an e x a c t sequence o f B-m odules, and over

A g,

then

M^g i s f r e e o v e r

a E-module o f f i n i t e

such th a t each f a c t o r p.

C B

t o p rove ( * )

i s fr e e over

A^,

Ng i s f r e e

To use t h i s , r e c a l l th a t

M

b e in g

ty p e , adm its a co m p o sitio n s e r ie s : (0 )

id e a l

L^

A^g.

= MQ C M1 C M2 C . . . M^+1

C Mh = M

i s isom orp h ic to

B/^

(BOURBAKE, A l g . Comm. , Ch. k, § 1 . 10 .

f o r th ese

and then i t

f o r some prim e

T h e r e fo r e i t

i s p roven f o r any

M.

s u f f ic e s

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

58

T h e re fo re we may assume Let

K be the q u o tie n t f i e l d

M = B,

of

and

A , and

B

L

i s an i n t e g r a l domain.

the q u o tie n t f i e l d

We s h a ll p ro ve ( * ) by in d u c tio n on the transcendence d egree over

K.

B

K;i t

th a t

B ^ K

n

of

B.

of

L

F i r s t , a p p ly N o e th e r 's n o r m a liz a tio n lemma t o the K -a lg e b ra f o llo w s th a t th e re e x i s t

a lth o u g h

n

elem en ts

f 1, . . . ,

i s i n t e g r a l o v e r the p o ly n o m ia l r i n g

B i s n o t n e c e s s a r ily i n t e g r a l o v e r

f

c B

K tf^ ...,

A [fj,...,

such

f n] .

Then

th e re

are

o n ly a f i n i t e number o f denom inators o c c u rrin g in th e r e l a t i o n s o f i n t e ­ g r a l dependence o f f o r some

th e g e n e ra to rs o f

(# )

Bf

Then

Bf

i s i n t e g r a l over

i s an Af [ f 1, . . . ,

can fin d

m

submodule

elem en ts

over K

[

f

f

o f Bf ,

Now

n] :

T h e r e fo r e ,

cm € B^

ty p e :

A ^ tf^ ...,

f*n J-

f n ]m — Bf — D — 0 .

is c le a r ly a fr e e

A^.-module, so i t

But, f i n a l l y , r e p la c in g D

Bf

c o n seq u en tly , we

g e n e r a tin g a f r e e

o f transcendence d egree

s u f f ic e s t o p ro ve

by the q u o tie n ts o f a s u f f i c i e n t l y

f i n e com p o sitio n s e r i e s , we a re reduced t o p r o v in g a lg e b r a s

.

such th a t the q u o tie n t i s a t o r s io n module

f n ]m f o r D.

A ^ C f^ ,..., f n ]

f n ]-m odule o f f i n i t e

c 1, . . . ,

0 -* Af [ f , , . . . ,

(* )

B

f € A,

le s s

( * ) fo r in t e g r a l A-

than no v e r

A. QED

3 Let

p

o

We a re l e f t w ith th e g e n e ra l c a s e ;

be the p r o je c t io n from

Pn x S 6m

As a f i r s t (* )

a coh eren t

S,

p* ( J (m ))

?

on

Pn x S.

and p u t: ’

s te p , we n o te :

th e re i s a of

=

to

fin ite

S such th a t

s e t o f l o c a l l y c lo s e d su bsets S = U Y^,

and such th a t i f

i t s reduced subscheme s tr u c tu r e ,

?

o^.

Immediate by 2° and th e d . c . c .

Y^

i s f l a t o v e r Y* . 1

“ *i P r o o f:

Y 1, . . . ,

Y^ i s g iv e n

f o r c lo s e d subsets o f

S.

From t h is we conclude s e v e r a l s im p lify in g f a c t s : (i)

t h e r e , i s a u n iform s € S,

mQ

H^C Pn g , 3rg (m ))

as in L e c tu re 7) and

such th a t s

if

(0 ),

® K( s)

m > mQ, then f o r a l l f o r i > 0 (n o ta tio n s

i s isom orp h ic t o

H°( PR g ,

y s (m ))P r o o f: schemes

Put to g e th e r ( * ) ;

§7, 2° p a r t ( i i )

Yi ; §7, 3 °, C o r o lla r y

and 3 °, p a r t ( i )

a p p lie d o v e r the base a p p lie d t o the in c lu s io n

Y ± C S. (ii)

O nly a f i n i t e number o f p o ly n o m ia ls p o ly n o m ia ls o f the sheaves

P .,,...,

J Q on the f i b r e s

P^ occur as H ilb e r t PR g

over

S.

FLATTENING STRATIFICATIONS F ix

m0

n o e t h e r ia n ).

as i n ( i ) ,

and l e t

g : T -► S

Suppose f i r s t o f a l l th a t Si:

be any base e x te n s io n

P x T g n Then by C o r o lla r y 2 in 3 ", L e c tu re 7, th e c a n o n ic a l map g * ( s m) i s an isom orphism , and

-► T

is

q *( J g (m )),

g*( 6 )

the p r o je c t io n ) .

5

59

is

fo r

l o c a l l y f r e e on

C o n v e rs e ly ,

suppose

S

then

S

q : Pn x T

^

i s f l a t over

have ” g . c . d .

fo r a l l T.

s tr a t ific a t io n ” :

,

and one can endow

W^j

the sum o f the sheaves o f id e a ls d e f in in g in g s t r a t i f i c a t i o n . fic a tio n fo r of a ll

y

Y^

m > mQ.

To be p r e c is e ,

fla t t e n in g s t r a t ific a t io n o f rank e . th a t,

Let fo r a l l

P^...,

Pk

Each f i n i t e

c lo s e d subscheme.

s

Let

U

Then I c la im

n

y (m )

m=m0 i Pi (m)

in t e r s e c t i o n i s ,

be in P. J

Yp

as ju s t e x p la in e d , a l o c a l l y

P j(m )

f o r th e n+1 v a lu e s o f

be the H i lb e r t p o ly n o m ia l o f

S in ce the h ig h e r cohom ology o f

Pp - P j

be the component o f the

becomes l o c a l l y f r e e o f

But, s e t - t h e o r e t i c a l l y , mn+n

Let

mn + n.

Y^m^

oo i

and

o f the f l a t t e n i n g s t r a t i f i c a t i o n s

le t

be th e H ilb e r t p o ly n o m ia ls o f ( i i ) .

i,

P r o o f:

By the

has an a s s o c ia te d f l a t t e n ­

onw hich

7

makes sen se:

Zy

and d e fin in g

What we have ju s t p roven i s th a t a f l a t t e n i n g s t r a t i ­

i s e s s e n t i a l l y the g . c . d .

fo r

W^j = Supp (Y ^ )

w ith a scheme s tru c tu re b y ta k in g

r e s u l t o f 1 °, each o f the co h eren t sheaves

But

Jg

(where is fla t ,

i s a ls o the union o f th e l o c a l l y c lo s e d su bsets

n Supp ( Z j ) ,

U

T.

g iv e n S = U Y± = U Z.

mn

T

g*( g )

then by C o r o lla r y 3 in 3 , L e c tu re 7, Now any two s t r a t i f i c a t i o n s o f

(T

i s f l a t over

m > m0

o

m > mQ:

i.e .,,

on

? s va n ish es by ( i i ) ,

= dim

® K (s )

has d e g re e a t most

n,

and

y

S

m on

between p

il y S

.

we have

= P1 (m)

n+1 z e r o e s :

. t h e r e fo r e i t

is

id e n t ic a lly zero. QED C on sequ en tly,

Zp

i s th e l i m i t o f a d escen d in g ch ain o f l o c a l l y

c lo s e d subschemes w it h f i x e d su p p ort, i . e . , f i x e d open s e t

U.

By the d . c . c .

Zp i s a c t u a lly a f i n i t e

5

over

S.

te rm in a te s and

in t e r s e c t i o n w hich makes sense.

I t i s now t r i v i a l th a t fo r

o f c lo s e d subschemes in a

f o r c lo s e d subschemes, i t

Z1, . . . ,

Zk

is a fla tte n in g s t r a t ific a t io n

60

LECTURES ON CURVES ON- AN ALGEBRAIC SURFACE An ob viou s s tre n g th e n in g o f the r e s u lt i s Let

C o r o lla r y :

where ?

i

f

X

S

t h is :

be a morphism w hich can be fa c t o r e d :

i s a c lo s e d im m ersion.

Let

d e fin e s a f l a t t e n i n g s t r a t i f i c a t i o n

5

be a co h eren t sh e a f on (Z i )

on

A n oth er im p ortan t consequence o f our method o f p r o o f i s s t r a t ific a t io n i)

(Z i )

can be in d exed by H ilb e r t p olyn o m ia ls

the induced sh ea f

J

o„

£s Pi ii)

if

on i

i

Pn * Zi>

^ j,

then

P1 ^ P j

.

X:

then

S. th a t the so th a t

has H ilb e r t p o ly n o m ia l

LECTURE 9 CARTIER DIVISORS 1°

We assume th a t

D e f i n i t i o n - P r o p o s it io n : X suoh th a t

X

i s a n o e th e ria n schemew ith s tru c tu re sh ea f

There i s a unique

sh ea f ( o f o^-m odules)

r (U , K^) = t o t a l q u o tie n t r i n g o f and f o r

U C V, P r o o f:

Spec ( R ) ,

th e r e s t r i c t i o n i s

b j ai

y

the n a tu r a l one.

and assume (a i /bi

" aj bj

is

R, p n o t a o - d i v i s o r i n i)

M u lt ip ly in g

R

0

R ( f £ )•

in

such th a t

a^

and

|

th a t

bi

Put

aj_bj = a j bi

c • b^

must go t o

* o, 0

in

t h is im p lie s th a t iii)

U =

Then we must fin d

aib^ - pa^ i s 0

by f ^

b - j,..., a ll

i.

R^f y

(fo r N »

But

in

a, R^

y

0, and a l l are in

Now say

(b ± )

c e R

and

i ) ,

R, and

N fV • c = 0.

S in ce

in R( f y

a ll i ) .

c • 91= ( o ) .

y

so

1€ ( f ^ . . . ,

f n) ,

b^ i s a n o n - 0 - d iv is o r i n

R( f

c

c = o.

But s in c e

R

i s n o e th e ria n , any

t a in s a non 0- d i v i s o r

p.

i s a c t u a lly a s e c t io n o f

a, e R,

b^

i s i n the i d e a l

bn e %.

i.e .,

a g re e on

R#

91 = {p € R | pa±

Then one checks th a t Then

Say

1 < i < n}

we can assume th a t a l l elem en ts

ii)

o^)

XL. = Spec \ a re g iv e n where U- , 1 < i < n, form i± i± U: i.e ., 1 e ( f ^ . . . , f R) . Suppose ai , b^ € y b^

n o t a o - d i v i s o r i n R^f U.j_ n U j, i . e . ,

r (U ,

E v e r y th in g i s e a s i l y reduced t o t h is p o in t :

and

a c o v e r in g o f

p €

on

f o r a f f i n e open U C X,

p * ai /bi

Now i t o^.

91

w ith t h is

p r o p e r ty con­

f o llo w s th a t p • aj_A>j_

o yer

U, hence f o r some

= a. QED

61

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

62

We m ention th a t

i s n o t alw ays qua s i- c o h e r e n t !

checks th a t the s ta lk s the s ta lk s

£x .

of

K^-

F i n a l l y , we can d e fin e

o f the sh ea f o f r in g s

K^,

t o be the subsheaf o f u n its

i.e .,

r (U , K^) = i n v e r t i b l e elem en ts o f N ote th a t

C

D e fin itio n : K^/o^ .

and

A ls o , one

a re ju s t th e t o t a l q u o tie n t r in g s o f

o^ C

r (U , K^)

.

.

A C a r t ie r d i v i s o r

D

on

X

i s a s e c t io n o v e r

More c o n c r e t e ly , a C a r t ie r d i v i s o r i s

X

of

g iv e n b y a c o l l e c t i o n o f

elem en ts Dx e ^ / ° x such t h a t , f o r a l l elem ent

x , th e re i s an open neighborhood

f € r (U , K^)

w hich in du ces

w i l l be c a lle d a l o c a l e q u a tio n o f in

£.

{f^ }

Dx D

fo r a l l

in

U:

U

x € U.

of

X, and an The elem en t f

I t i s unique up t o a u n it

A C a r t ie r d i v i s o r can be d eterm in ed by s p e c ify in g l o c a l e q u a tio n s w ith r e s p e c t t o an open c o v e r in g

u n it in

(U ^ ),

so lo n g as

is a

n U^ . Note th a t th e s e t o f a l l C a r t ie r d iv is o r s from s a group.

A lth ou gh

t h is law comes from m u lt ip ly in g l o c a l e q u a tio n s , we f o l l o w h a llo w ed con­ v e n tio n and w r it e i t a d d i t i v e l y : +1 f 1 • f-^o f l o c a l e q u a tio n s .

i.e .,

as

A s s o c ia te d t o a C a r t ie r d i v i s o r

D

£x (D) w hich i s an i n v e r t i b l e

sh ea f o f

f

i s the elem ent o f

i s a coh eren t subsheaf:

£x -m odules.

U,

Namely, f o r a l l

x,

p u t:

’ 2.X C

induced b y a l o c a l e q u a tio n

T h is i s c l e a r l y independent o f th e c h o ic e o f e q u a tio n i n

f o r th e com bination

C

I ° x ( D> ] x = f x where

D- + Dp

f,

and, i f

f

f

of

D.

i s a lo c a l

then

—X I U m u lt, by

’ 2 x^

'u

f _1 i s an isomorphism o f sheaves o f

£x -m odules.

I t i s n o t hard t o check th a t t h is a c t u a lly g iv e s an isomorphism between the s e t o f C a r t ie r d i v i s o r s on h eren t subsheaves o f

K^.

X, and the s e t o f i n v e r t i b l e c o ­

CARTIER DIVISORS D e fin itio n :

A C a r t ie r d i v i s o r D

i) or

its

lo c a l

i i ) ox C £X (D)

or

i i i ) £X (-D )

We s h a ll w r i t e :

is e ffe c t iv e i f

e q u a tio n s f

a re

D > 0 t o mean

0

D

is e ffe c tiv e .

o^

Suppose

£X (-D ) -

£x “ ►

on th e t o p o l o g i c a l space w hich i s

la n gu age, we s h a ll a ls o c a l l t h is c lo s e d subscheme f

in

i s an e f f e c

— 0

o^, one o b ta in s a c lo s e d subscheme o f

subscheme d eterm in es i t s

D

denote th e c o k e r n e l:

one ta k es th e s tru c tu re sh e a f

l o c a l e q u a tio n s

£x ,

i s a sh eaf o f id e a ls .

(* )

the support o f

e q u iv a le n t ly :

s e c tio n s o f

C

t i v e C a r t ie r d i v i s o r , and l e t

If

63

s h e a f o f id e a ls ox ( v i a £X(-D )

term in ed by the c lo s e d subscheme

D

o ^ - D ),

= f

X:

D.

By abuse o f

Sin ce t h is c lo s e d

w hich in tu rn determ in e

* £x ) ,

th e C a r t ie r d i v i s o r

D> i s

and our co n fu s io n should n o t be dan­

gerou s . M oreover, when in

D > o , the image

s o f the s e c tio n

r (X , £x ( D ) ) w i l l be c a lle d the g lo b a l

e q u a tio n o f

D.

1e r (X , £x ) In f a c t , i f we

le t £X (D) be any isom orphism o f m odules,

ox

cp(s)

M o reo ver, in the e x a c t sequence

i s a l o c a l e q u a tio n f o r

(* ),

th e in c lu s io n o f

can be in t e r p r e t e d as te n s o r in g w ith A C a r t ie r d i v i s o r D e fin itio n : x € X

The support o f

a t w hich

D e fin itio n :

D

1

is

D

£X(-D )

P ic

x. £x

s.

d eterm in es even more th in g s : D

is

the c lo s e d subset c o n s is t in g o f

those

n o t a l o c a l e q u a tio n .

The d i v i s o r c la s s a s s o c ia te d t o the C a r t ie r d i v i s o r

the elem ent o f

at

in

(X )

D

is

o b ta in e d by th e co-boundary: H °(X , K */ o *)

—

H1 (X , £ * ) II P ic

(X )

,

v i a th e e x a c t sequence: (# ) c

° -

One checks im m ed ia tely th e i n v e r t i b l e

sh e a f

oj -

K* -

th a t t h is elem en t o f P ic

D e fin it io n :

Two C a r t ie r d i v i s o r s e q u iv a le n t ly ,

i)

(X ) i s ,

in f a c t , g iv e n

by

£X( D ) .

D1 = D2) i f ,

ii)

K£/o* - 0

£X (D1) & £x (D2) ,

as

the d i v i s o r c la s s o f

D1, D2

a re l i n e a r l y e q u iv a le n t (w r it t e n

£x ~modules, D1

eq u a ls the d i v i s o r c la s s o f

Dg ,

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE iii)

th e re i s an

f e r (X , K^) f-o ^ D ,)

such th a t =

o

o % D e fin itio n :

If

f € r (X , K ^ ),

•

then the C a r t ie r d i v i s o r w ith

l o c a l e q u a tio n everyw here w i l l be denoted

(f).

(# )q , o n e

p r i n c i p a l , and by use o f the e x a c t sequence D1 = D2 i f and o n ly i f D1 = D2 + ( f ) , N e x t, suppose an i n v e r t i b l e a l l e f f e c t i v e C a r t ie r d i v is o r s say, lo o k f o r isomorphisms

D

sh e a f

L

f

as i t s

Such d i v i s o r s a re c a lle d see s :

f o r some

f € r (X , K^) .

i s g iv e n —c o n s id e r the s e t o f

whose d i v i s o r c la s s i s

L.

That i s t o

a: L 9

/

/

/

1°

ox C ox (D) L e t t in g

be the co m p osition in the diagram , one sees c o n v e r s e ly th a t

cp

f o r e v e r y i n j e c t i v e homomorphism such th a t

C Kx

th e re i s a unique C a r t ie r d i v i s o r D

cp,

a

o f £X (D)

can be determ in ed , f o r exam ple, b y l e t t i n g

s = q > 0 ),

cp

exten d s t o an isomorphism

and L .

Thus

D

and ch oosin g l o c a l

isom orphism s: L ^ Then the image o f we c a l l cp

s

s € r (X , L )

is in je c t iv e

in

'u ±

r(U ^ , £x )

•

i s a l o c a l e q u a tio n f o r

a g lo b a l eq u a tio n _£ o r

D. Note

correspon ds t o the f a c t th a t

D.

As above,

th a t th e f a c t th a t

si s n o t a o - d i v i s o r .

The

above re a s o n in g le a d s t o : P r o p o s it io n :

If

L

i s an i n v e r t i b l e s h e a f, then th e re i s a n a tu r a l i s o ­

morphism: e f f e c t i v e C a r t ie r d i v i s o r s I D s .t. £x (D) s L J

Example: X

Let

X = P r o j k [X Q, . . . ,

c a r r ie s the sh ea f

£x ( 1 ),

X^,

s e c tio n s s € r (X , L ) , n ot o - d i v i s o r s , modulo s ~ a • s, f o r a € r (X , £ j ) k

a fie ld .

Then as in L e c tu re 5,

and th e re a re homomorphisms:

j v e c t o r space o f homogeneous \ I form s i n X , . . . , X o f d eg ree dj o n T h e r e fo r e , each form

F (X Q, . . . ,

o f an e f f e c t i v e C a r t ie r d i v i s o r d

*

Xn) o f d egree D C X

d

such th a t

i s c a lle d th e h y p ersu rfa ce w ith e q u a tio n

h y p e r p la n e ).

^ f aw

F,

is

the g lo b a l e q u a tio n

£X(D) “ £^(6 .). (o r , i f

d = 1,

T h is the

CARTIER DIVISORS 2° If

C a r t ie r d i v i s o r s a re c l o s e l y r e la t e d t o the con cep t o f d epth . where d ep th ( o j hence —Zt = o ,* then K —Zi = o—Z e v e r y C a r t ie r d i v i s o r i s t r i v i a l i n a neighborhood

z € X i s a p o in t = t 1)*

of

z.

The rem arkable th in g i s

th a t C a r t ie r d iv is o r s a re determ ined by

t h e i r e q u a tio n s a t p o in ts o f dep th P r o p o s it io n :

Let

X.

Then D1 = D2

a re

equal fo r

D1

and

X

be a n o e th e ria n scheme, x where

It

D2

1 :

i f and o n ly i f

a ll

P r o o f: of

s u f f i c e s t o p ro ve th a t th e im ages

a re eq u a l i n a l l s ta lk s

(a .j), ( 0)

^

then i n

fP

I1 = I2

a s s o c ia te d t o

/a, * ( © ) '

( 0)

,

I1

0,

a1

t h is red u ces t o p r o v in g :

then

I1 = I2

( 0 )p

I 2.

if

And i f

^

depth

*

I.,(G

=

o f d ep th 1 . f o r a l l prim e

i s a s s o c ia te d to

i s a n o n -O -d iv is o r such th a t

a re 0 - d iv is o r s : i . e . ,

r

( Di ) x B ut, m u lt ip ly in g

I 1( 0 ) p = l 2( 0

if or

(K * / o * )x .

o ,

fo r a l l lo c a liz a tio n s

But c e r t a i n l y

(K * / o * )x

I 1, I 2, g e n e ra te d by n o n - 0 - d iv is o r s ,

in a l o c a l n o e th e ria n r i n g

id e a ls

two C - d iv is o r s on

d ep th (_o ) = 1 .

G iven two p r in c ip a l id e a ls I 2( 0 )p

D1, D2

t h e ir im ages i n th e s ta lk s

b o th by a s u ita b le n o n -O -d iv is o r in (* )

65

I1 =

a l l n o n -u n its i n

( 0 ) = 1. p

QED In a v e r y s im ila r way, i t i s e f f e c t i v e i f and o n ly i f

it

can be p roved th a t a C a r t ie r d i v i s o r

i s e f f e c t i v e a t a l l p o in ts

D

x , where depth

(o x ) = 1 . C o r o lla r y : £x

Let

X be a norm al n o e th e ria n scheme, i . e . ,

a re i n t e g r a l l y c lo s e d domains.

eq u a l i f and o n ly i f P r o o f:

th e y a re e q u a l a t

By th e p r in c ip a l

K r u ll dim ension

> 2

has d ep th

a l l lo c a l

Then two C a r t i e r - d i v i s o r s a l l p o in ts

x

r in g s

D1, D2 a re

o f codim ension 1,

i d e a l theorem , anorm al l o c a l r in g o f > 2. QED

Now assume f o r th e r e s t o f 2* th a t n o e th e ria n scheme. then

is

If

K

is

th e s t a lk o f

X

i s an ir r e d u c ib le normal

£x

a t the g e n e r ic p o in t o f X,

sim p ly th e co n sta n t s h e a f: r (U , Kx ) = K,

In c id e n ta lly ,

a ll

t h is p ro v e s im m ed ia tely th a t

th e e x a c t sequence

( # ) q ( 1#) :

U . 1

H (X , Kx ) = ( 0 ) ,

every in v e r t ib le

s h ea f

£

on

hence by X

d i v i s o r c la s s o f some C a r t i e r - d i v i s o r . D e fin itio n :

A W e il d i v i s o r on

X

i s a fo rm a l sum

n

I

r i E±

i= l where

E 1, . . . ,

En a re c lo s e d ir r e d u c ib le

su bsets o f codim ension 1.

is

the

66

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE If,

fo r a l l

x € Xo f codim ension 1, r (U , z T ) = r (0 ) x 1 Z

then one checks th a t

a W e il d i v i s o r i s

we d e fin e a sh ea f

Zx by:

x 4 u x e U

if if

the same th in g as a s e c tio n

of

the

sh eaf

1 zxx

x o f codim

*

Now th e re i s a c a n o n ic a l e x a c t sequence:

0 - O* - k £

(#)w w

—a

e

—

f e r (U , K^) = K *,

Namely, g iv e n

^

z

x o f codim 1

.

x

d e fin e i t s image t o b e :

ordx ( f ) • {50

,

X€U x o f codim 1 where

ordx ( f )

Spec (R ) .

i s th e o rd e r o f f

Then l e t

^

(t j = tpy

[= Rn(^*R )

IE i= l N ote th a t i f R:

(t

)

r.

s^

U=

(t ) *>n n

n ...n

.

P

n

in

In o th e r w ords, say

a re m inim al prim e i d e a l s , and

fp

power o f

x.

= g/h, where g , h € R, and l e t (S ,) ( s 2) (s ) (g ) =p1 n fP2 n . . .n pn , s± > o (h )

where the

at

f

^si~ ^i^

f ° l osure

= t^ f o r a l l

i s th e t ttl "s y m b o lic "

Then th e image o f f

i,

p o in t g iv e n by then

(g )

is :

p^} .

= ( h ) , hence

f

i s a u n it

t h is shows th a t ( # ) y i s e x a c t. P u ttin g ( # ) q and ( # ) y t o g e t h e r , we o b ta in an in c lu s io n —*/ —* c

®x zx

»

hence th e group o f C a r t ie r d i v i s o r s i s embedded i n the group o f W e il d i ­ v is o r s . above: in

T h is i s , fo r i f

£x ,

ju s t an in t e r p r e t a t io n o f th e C o r o lla r y ju s t

has codim ension 1, and ( tt)

then the s t a lk o f a C a r t ie r d i v i s o r a t

o f the form

7rr ,

W e il d i v i s o r i s P r o p o s it io n : v is o r s i f

in f a c t ,

x e X

f o r a w e l l d eterm in ed in t e g e r then ju s t the sum o v e r

x

of

r

i s the maximal i d e a l x

has a l o c a l eq u a tio n

r.

The co rresp o n d in g

• {5 0 .

The group o f C a r t ie r d i v i s o r s eq u a ls th e group o f W e il d i ­

and o n ly i f a l l l o c a l r in g s

r e g u la r scheme.

ox

a re UFD's;

e .g .,

if

X

is a

CARTIER DIVISORS P r o o f:

The two ty p e s o f d i v i s o r s a re eq u a l i f

homomorphism o f s t a lk s in

s u r je c tiv e .

and o n ly i f

the

(#)y**

(5 j)v A y is

67

But t h is i s K*

-

0

r

i x o f codim 1

zx

Jy

sim p ly: -►

©

Z

* c Sr m inim al prim es a s s ig n in g t o

fp . i.e .,

T h is i s

f

* g/h

th e d if f e r e n c e o f the o rd e rs o f

s u r je c t iv e i f

i f and o n ly i f

ox

and o n ly i f e v e r y i s a UFD.

p C £x

g

and

h

at a ll

i s a p r i n c ip a l i d e a l :

LECTURE 10 FUNCTORIAL PROPERTIES OF EFFECTIVE CARTIER DIVISORS 1°

The s im p le s t o p e r a tio n t o p erform w ith C a r t ie r d iv is o r s i s

take in v e r s e im ages: and say

D

g

say

X ------►Y

i s an e f f e c t i v e C - d iv is o r on

Y.

g * (D )

ought t o mean:

F ix an open c o v e r in g

t io n s

f^

U^,

fo r

D

in

where

be d e fin e d by l o c a l eq u a tio n s However,

to

i s a morphism o f n o e th e ria n schemes, Then i t {U^}

f^ e r(U .p

g * (fi )

i s q u it e c le a r what

of

Oy) .

Y

Then

and l o c a l equa­ g *(D )

i n the open c o v e r in g

g * (f^ )

can

be a o - d i v i s o r , even

fo r

a ll

x € A (X ),

0.

should

g " 1(U ^ ).

The b e s t th in g

is to as­

sume: (* ) Then

g * (f^ )

is

P r o o f: Then l e t

x T

th e s e c t io n

4

g (x )

n o t a o - d i v i s o r , and Suppose

a * g * (f^ )

g *(D )

=

0,

be th e g e n e r ic p o in t o f a

Supp (D)

of

a^

.

makes sense.

where

a e c>x ,

and

x e

X.

some component o f the support o f

(d e fin e d n ear

x) :

We maytake

x ' e Spec (o ) C X . Then o „ , has depth o s in c e the induced elem ent a ’ e o f i s k i l l e d —x e —x by a power o f the maximal i d e a l mx , ( c f . L e c tu re 8, 2 ) , and s in ce a ' ^ 0.

But then

x f e A (X ), f^

u n it a t

T h e r e fo r e , i n 4 £x ,:

x’.

fo r D

hence

c a l e q u a tio n

g ( x ')

i s a u n it a t

4 Supp ( D ) .

g ( x ') ;

a ' = [a 1 • g * (fi )]

T h e r e fo r e , the l o ­

t h e r e fo r e

*

=

g * (f^ )

is a

0 .

T h is c o n t r a d ic t io n p ro ves the r e s u l t . N ote th a t i f then f o r a l l

g

x e A (X ),

is fla t , g (x )

(* )

e A (Y ),

i s a u tom a tic. For i f (L e c tu r e 6 ),

hence

g g (x )

is f l a t , i s n o t in

the support o f any C - d iv is o r (L e c tu r e 8, 2 ) . 2 image

o

S *(D )

A more i n t e r e s t i n g q u e s tio n i s o f an e f f e c t i v e

when can one d e fin e a d i r e c t

C - d iv is o r

D

on

g is fin ite

and f l a t .

t r e a t the "e le m e n ta ry " ca se:

69

X. In t h is s e c t io n , we

70

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

Then

g*

sin ce i t Y,

can be d e fin e d by Norms!

is

l o c a l on

and l e t

Y:

on

Y , i f we take U a re then s e t

Theproblem i s e s s e n t i a l l y a lg e b r a ic ,

U = Spec

g ~ 1 (U) = Spec (B ) .

n i t e typ e as A-m odule. We

le t

Then

(A ) B

M oreover, s in c e

g# (o ^ ) i s a l o c a l l y f r e e

s u f f i c i e n t l y sm a ll,

up

be an open a f f i n e subset o f

i s an A - a lg e b r a , w hich i s

of f i ­

sh ea f

B i s a f r e e A-module to o .

f o r norms:

if

p € B,

le t

if

b 1, . . . ,

b

Tpi B -► B

be m u lt ip lic a t io n by

a re a b a s is o f

B

o v e r A,

p.

le t

n W

=

Z ai j b j • J-1

Then: Mn(p) = d e t ( aj_j) T h is i s n a t u r a lly independent o f th e b a s is

.

b^, and has th e obviou s p ro p ­

e r tie s : l®n( 01 • P2) = TBn( P ,) Nm(a)

=

a11,

• Mn( P2)

a e

if

A .

A lth ou gh the norm i s n o t alw ays a produ ct o f

p

and i t s

co n ju ­

g a te s , a t le a s t one has: (* )

fo r a l l P r o o f:

p o ly n o m ia l o f P (P )

=

p,

th e re i s a

Let T^.

P (X )

p’

such th a t

Nm(p) = p * p ’

= d e t (X • i d e n t i t y - T^)

be the c h a r a c t e r is t ic

Then (C a y le y -H a m ilto n theorem ) P (T ^ )

P ( T p ) ( l ) = 0,

o r , w r i t i n g out

.

= 0,

hence

P:

pn + a 1pn_1 + . . . + a n-1 • p + lta(p) = o. QED One a ls o has the im p o rta n t: (* * )

If

p e B

P r o o f: Lemma A : Let

x

then

Nm( p) i s n o t a o - d i v i s o r .

We use a sim ple g e n e r a l f a c t : g

Let

e X.

i s not a o - d iv is o r ,

X ---- ►Y I f g (x )

be a f i n i t e has dep th

f l a t morphism o f n o e th e ria n schemes. 0,

then x

has depth

0,

and

con­

v e r s e ly . P r o o f:

I f d ep th

whose a n n ih ila t e r i s g * : £ g (x ) fin ite , mg (x )

£x mg ( x )

* £x k i H s

g (x )

=

i s i n j e c t i v e and * £x

0, then th e re e x is t s a e

mg ( x ) > t*16 maximal i d e a l . g * (a )

e ox

Sin ce

i s n o t 0.

i s P rim ary f o r the maximal i d e a l

S * (a ) ,

the d ep th o f

x

is

£ g (x ) ,a

^ 0,

gi s f l a t , Sin ce

mx :

g

is

sin ce

0. The con verse was

proven in L e c tu re 6.

QED

FUNCTORIAL PROPERTIES OF EFFECTIVE CARTIER DIVISORS R e tu rn in g t o

B/A:

i s a prim e i d e a l » C A i s a o - d iv is o r in

Suppose

Nin( p)

i s a o - d iv is o r .

such th a t d epth ( A * ) = 0, A^

[i.e .,

le t

by

Bv = B

tio n s

A. Then

have d ep th

B

is a

Nm

i s m u ltip lic a tiv e ,

Nm( p)

th a t Nin( P)

R ep la ce

A

*be a m inim al

by

Av

and

s e m i- lo c a l r i n g a l l o f whose l o c a l i z a ­

0 b y th e lemma.

in none o f th e maximal i d e a ls o f

Then th e re

and such

a • Nm( p) = o, and l e t

prim e i d e a l c o n ta in in g the a n n ih ila t e r o f a ] . B

71

Then i f B, i . e . ,

p p

i s not a o - d iv is o r , i s a u n it i n

B.

p is

S ince

i s a u n it t o o , w hich c o n tr a d ic ts our a s ­

sumption. To a p p ly th e norm t o the d e f i n i t i o n o f Lemma B : le t

L

[U^}

Let

X

►Y

of

Y

such th a t

=

g*,

we need:

be a f i n i t e morphism o f n o e th e ria n schemes, and

be an i n v e r t i b l e

P r o o f: B

g

sh e a f on L

F or a l l y e Y , S in ce g

X.

Then th e re e x is t s an open c o v e r in g

i s isom orp h ic t o

is

o^

i n each open s e t

lo o k a t the module

fin ite ,

B

M

= g * (L )y

g ” 1 (U^) . o ver

i s a s e m i- lo c a l r in g , and i f

31

i s i t s r a d ic a l, B/ sT s

®

K (x )

.

x over y T h e r e fo r e ,

M/si • M

is

c e r t a i n l y f r e e o f rank 1 :

ran k 1 o v e r B ( c f .

BOURBAKI, A l g . Comm. , Ch. I I ,

be a b a s is o f

th en ,

M;

an open n eighborhood

U1

hence

M

§3, P rop .

jx

i s induced b y a s e c t io n

of

y.

is fr e e o f

5 ).

\± o f

Let

iiy

g * (L )

in

\± d e fin e s a homo­

M u lt ip lic a t io n b y

morphism: S* ( in

.

at U.J of

^

* S* ( h)

The k e r n e l andc o k e rn e l a re co h eren t sheaves

y a re o f y. o^

( 0 ) : t h e r e fo r e , b o th a re (0 ) Then in

g

(U 2) ,

on

Y

whose s ta lk s

in a whole neighborhood

m u lt ip lic a t io n by

UQC

n g iv e s an isomorphism

and L. QED Now in our ca s e , we a re g iv e n an e f f e c t i v e C - d iv is o r

By

the lemma, th e re i s an open a f f i n e

such th a t

D

is

p r in c ip a l in

g “ (U^)

c o v e r in g

U. =

D

on

X:

Spec (A . )

= Spec ( B ^ ) . T h e r e fo r e

D

o fY

i s de­

f in e d by an e q u a tio n p. € B . , fo r a l l i , p. n o t a O - d iv is o r . —1 —1 checks th a t p^ • p^ i s a u n it in r ( g (U^n U j ) , o_x ) , hence

One

Nm( p^)• N m (P j)“ 1

s e c tio n s

Nm(Pi )

i s a u n it

d e fin e a C - d iv is o r

3°

in

r(U ^ n U j,

Rem arkably, the d i r e c t image

much more g e n e r a l c a se:

2

o f ca s e s , in each o f w hich

O y ).

T h e r e fo r e , the

g * (D ) .

g *(D )

can be d e fin e d in a v e r y

i s r e a l l y ju s t "ca se o " in an i n f i n i t e g *(D )

set

can be d e fin e d , bu t r e q u ir in g , i n each

s u c c e s s iv e ca s e , the com putation o f one more d eterm in a n t, among o th e r

72

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

th in g s .

We have in mind the f o llo w in g s it u a t io n : Pn X Y 0 X 0 V g

So

YOU where (a )

X

(b )

i s a c lo s e d subscheme o f

V = g “ 1(U ),

(c )

gQ

(d )

g

(e )

a l l p o in ts

Pn x Y ,

U

gQ i s the r e s t r i c t i o n o f

is

open in

Y,

g,

is fin it e , is

of fin ite

Tor-d im en sion ,

y e Y,

where

Oy

has d epth

o

or

1,

a re in

Then in t h is s it u a t io n th e re i s a n a tu r a l d e f i n i t i o n o f g * ( D ) . Mumford, G eom etric In v a r ia n t T h e o ry , Ch. a ls o f l a t ,

o

C - d iv is o r .

Suppose

typ e o f n o e th e ria n schemes.

-*(D ly) '

X

►Y

P r o p o s it io n - D e fin it io n :

D

i) or

ii)

is fla t

fo r a l l

An e f f e c t i v e C - d iv is o r Y

if

iii)

(ii)

c a l on

X

and

O - d iv is o r . t o (0 )

s a tis fy

:

be a prim e i d e a l .

Since

t h e r e fo r e a l l prim e id e a l s

.

A/p

F

i s n ot

A/p

of

B

a non-

/A,

B/p • B

is f l a t

C B/p

• B

a s s o c ia te d t o ( o ) ,

lo ­

F € B

a s s o c ia te d i.e .,

con­

i s an i n t e g r a l domain ( t h i s i s Example 1,

T h e r e fo r e , a l l such

correspon ds t o i i i ) '*

x

y = f (x ),

B, a f l a t A - a lg e b r a , and

In o th e r w ords, a l l prim e id e a ls

* n A = p

B [q u o tie n t f i e l d y

at

a re o b v io u s ly e q u iv a le n t. To p ro ve them

then one has

i t s e l f s in c e

L e c tu re 6) .

D

n supp (D) = 0 .

c o n tr a c t to prim e id e a ls in

t r a c t (0 )

of

K ( y ) , where “Y

and ( i i i )

p CA

A/p

s a id t o be a

pass t o the a lg e b r a ic setu p , sin ce the problem i s

Y:

Let

i s f l a t over

F

£x

y e Y, A ( f _1 ( y ) )

P r o o f:

is

/Y,

x € X, the l o c a l eq u a tio n

fo r a l l

e q u iv a le n t t o ( i ) ,

D C X

f.

e q u iv a le n t ly :

a z e r o - d iv is o r in th e r i n g

if

i s a f l a t morphism o f f i n i t e

The q u e s tio n i s , when should a d i v i s o r

be re g a rd e d as a fa m ily o f C - d iv is o r s on the v a r io u s f i b r e s o f

r e la t iv e d iv is o r over

or

(C f.

f a c t , i f gQi s

In t h is s e c t io n , I want t o d e fin e the con cept o f a r e l a t i v e

(e ffe c tiv e ) D C X

In

g *(D ) i s u n iq u e ly d eterm in ed by the req u irem en t: g *(D ) Ip- -

4

5, § 3 .)

*

v C B

a s s o c ia te d t o

are a s s o c ia te d t o (0 )

p • B in

A/p ] , i . e . , such v correspon d t o x € A ( f _1 ( y ) ) p . T h e r e fo r e , h yp o th e s is ( i i i ) a s s e r t s :

i s n o t i n any a s s o c ia te d prim e i d e a l o f

f o r any prim e i d e a l

p CA .

p • B,

FTJNCTORIAL PROPERTIES OF EFFECTIVE CARTIER DIVISORS To p ro ve t h is i s

e q u iv a le n t t o the f la t n e s s

th a t f la t n e s s i s

e q u iv a le n t t o :

Tor^ (B/P-B, a l l prim e id e a ls § 4 ).

ip

C A,

A/p )

of

B/F • B

=

o ver

73 A,

r e c a ll

(0) ,

( t h i s i s easy— c f . BOURBAKE, Comm. A l g . , Ch. I ,

But u s in g :

Tor^ ^

A/p ^

and the fla t n e s s o f

x/t? ) -♦ B/p • B

TorA

B o v e r A,

the v a n is h in g o f t h is

* B/p B Tor

i s e q u iv a le n t

to ( i i i ) * . QED The im p ortan t p o in t c o n cern in g r e l a t i v e C a r t ie r d iv is o r s i s

t h is :

g iv e n a f i b r e p rod u ct s it u a t io n :

and an e f f e c t i v e C - d iv is o r alw ays d e fin e d . x* e A ( X ')

D

in

X, r e l a t i v e t o

A (f’

( y 1) ) ,

if

= f -1 (y ) Spec

y = g ( y r) «

g f ( x ! ) € A ( f ~ 1( y ) ) .

T h e r e fo r e

f ’

—

1 ( y 1)

y'

x H (y)

T h is im p lie s th a t

= f '( x ) . Spec

is

And

M (y ')

i s f l a t over

f

“ 1 (y ),

hence

T h e r e fo r e g ’ ( x ’ ) 4 Supp (D) g f* (D )

In p a r t ic u la r ,

i s d e fin e d .

one can take

(C f.

. 1 ° ).

Y T = Spec H (y)

and one o b ta in s a fa m ily o f C - d iv is o r s on the f i b r e s r e q u ir e d I

g '* ( D )

F o r, by th e remarks a t the end o f L e c tu re 6, a p o in t

i s a ls o in f ' -1 ( y ')

where

_f, then

f o r v a r io u s f ~ 1 (y )

of

y e Y, f ^as

LECTURE 11 BACK TO THE CLASSICAL CASE A f t e r spending so lo n g in the a r id g e n e r a lit y o f a r b it r a r y n o e th e r­ ia n schemes we r e tu r n t o our p ro p e r cu rves

on a g iv e n s u r fa c e .

w ork in g o v e r a f i e l d F ix ,

program^-to i n v e s t ig a t e th e s e t o f

In t h is le c t u r e , we sim p ly s e t the sta ge f o r

k, r e c a l l i n g w ith o u t p r o o f some o f the b a s ic f a c t s :

once and f o r a l l ,

an a l g e b r a i c a l l y c lo s e d f i e l d /k i s a scheme

typ e o v e r

h e n c e fo r th , w i l l be a lg e b r a ic

k.

A l l schemes,

X

k.

R e c a l l , an a lg e b r a ic scheme

o f fin ite ,

schemes, and a l l fu n c to r s w i l l be fu n c to r s on the c a te g o r y o f a lg e b r a ic schemes. and i r r e d u c ib le P r o j k [X Q, . . . , (I.)

R e c a ll, a v a r i e t y

scheme/k. X^,

From

now on,

(not- P r o j Z[XQ, . . . ,

(* )

If

X

2, p.

Xn l ) .

X,

scheme, th e re i s an in t e g e r

D e fin it io n : It

If

X

can be

X:

i s any scheme, l e t

thus

shown t h a t if

dim

dim (X )

n

D e fin itio n :

A scheme

X

be the maximum o f the

(X )

i s a ls o th e c o h o m o lo g ic a l dimen­

H ^ X , J ) = ( 0)

i > dim X,

f o r a l l sheaves

5

197) •

is p r o je c t iv e

i t i s isom orp h ic t o a c lo s e d (r e s p .

(r e s p . q u a s i - p r o je c t iv e )

l o c a l l y c lo s e d )

subscheme o f

n ).

D e fin itio n :

An i n v e r t i b l e

sh e a f

L

on a scheme

th e re e x i s t s an im m ersion X ( f o r some

=

X.

GODEMENT, T h e o rie des fa is c e a u x , p.

( f o r some

H (x )/ k

x € X.

dim ensions o f th e components o f s io n o f

n,

such th a t

K r u ll dim (o x ) + tr a n s . deg. fo r a l l

(II.)

w i l l den ote

193):

i s an ir r e d u c ib le

the dim ension o f

if

i s a reduced

p

R e c a l l a ls o the main r e s u l t o f dim ension th e o ry in t h is case ( c f .

ZARISKI-SAMUEL, v o l .

(c f.

/k

n)

such th a t

q > * (o (i))

-

=L 75

pn .

X

i s v e r y ample i f

PR

76

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE There are several important remarks to make about this concept: a)

Suppose more generally that

L s cp*(_o(l))

for any morphism

cp: X -*■ Pn at all. Then the induced sections s^ = cp*(X^) of L span L. Conversely, if L is spanned by its global sections, one can choose a finite set sQ , s.,,..., s.,,..., ssof of sections sections which which span spanL. L.Then Then(L;(L; sQ,sQ, ..., sn)

defines an an X-valued X-valued point point of of Pn Pn,, i.e., i.e., a morphism

such that cp*(o(i)) = L . by its global sections.

cp: X -*• Pn ,

In particular, a very ample sheaf is spanned

b) Suppose H°(X, L) is finite-dimensional, e.g., suppose X is a projective scheme. Then if L is spanned by its sections, there is s q>*(£0)): q>*(_o( l)) : namely a nearly canonical morphism cp: X -► Pn such that L ^ namely, sQ, s1,..., sn of These cannot all vanish take a basis sQ, of H°(X, H°(X, L) L) .. These cannot all vanish at any one point, so (L; (L; sQ sQ,..., ,..., sn sn)) defines defines such such aa cp. More functorially, functorially this defines a morphism: tp:
°x)

n1 ( x >

® L _1 )

h1 ^x » II

(0 ) Since

dim H1(X , o^)

= pa ,

.

t h is i s a c o n t r a d ic t io n . QED

A fu r t h e r developm ent o f the th e o ry shows th a t s ta n t

rig,

sh ea f

L

depending o n ly on

Am. J .

Math. , 1964).

i s a t l e a s t nQ,

o n ly i f d eg (L ) > 0 . theorem th a t

X, such th a t when th e d e g re e o f the i n v e r t i b l e

th en

°o

L

i s v e ry ample ( c f .

MATSUSAKA-MUMFORD,

T h is g iv e s the e le g a n t C o r o lla r y r

L

i s ample i f and

[T o show th a t Mdeg L > 0M i s n e c e s s a ry , use S e r r e 's

ML am ple” im p lie s

X (L n) -♦*

th e re i s an oth er con­

as n -*■

+oo,

H1(L n) = ( 0 ) ,

f o r la r g e

n, hence

hence, b y Theorem 1 deg (L ) > o . ]

BACK TO THE CLASSICAL CASE F in a lly ,

81

th e re i s a t h ir d p a r t o f th e Riemann-Roch theorem w hich we

s h a ll use i n the

n e x t l e c t u r e . T h is i s a r e s u l t w hich en a b les one

th e s h e a f

some c a s e s :

a>p i n

THEOREM 3:

Let

F

be a n o n -s in g u la r p r o j e c t i v e s u rfa c e .

Then th e re i s a c a n o n ic a l i n v e r t i b l e th e fo llo w in g p r o p e r ty : v is o r .

Then

t o compute

Let

D C

s h ea f

F

ft

on

F

w ith

be any e f f e c t i v e d i ­

Di s a cu rve and a>P s* [ft Op(D) ] Op .

Example:

If

F = p2 ,

th en , as i s

D C Poi s a p lan e curve 3

w ell-k n ow n ,

o f d e g ree d,

i.e .,

= o (- 3 ).

ft

o p (D)

Then suppose

= o (d ) .

Then Theorem

2

t e l l s us th a t cop a o (d - 3 ) ®

.

F or exam ple, i f d = 3, th en , Op)

a re d u a l, hence X (^ )

= 0

pa ( D) = 1 Such cu rves a re known as e l l i p t i c

.

cu rves when

D

i s n o n - s in g u la r .

LECTURE 12 THE OVER-ALL CLASSIFICATION OF CURVES ON SURFACES We now tu rn our a t t e n t io n t o geom etry on a f i x e d p r o j e c t i v e and non­ s in g u la r s u r fa c e ,

F.

On

F

we have d i v i s o r s

(W e il or C a r t ie r , i t makes no

d i f f e r e n c e ) , and th e group o f d i v i s o r c la s s e s

P ic

(F ).

Among d i v i s o r s ,

e f f e c t i v e d i v i s o r s w i l l be r e f e r r e d t o sim p ly as cu rv e s :

the

th ese a re now 1-

d im en sio n a l c lo s e d subschemes, but th ey a re n o t n e c e s s a r ily reduced or i r r e ­ d u c ib le . 1°

Let

D CF

be a c u rve.

U n lik e the case o f e f f e c t i v e

d i v is o r s

on cu rves th em selves, one cannot count the number o f p o in ts in the support and c a l l i t can

th e d e g re e ,

s in c e th e support i s p o s i t i v e d im en sio n a l,

do i n th e way o f co u n tin g i s Let

D1 , D2

be two cu rves in

F

such th a t

dim (Supp (D 1) n Supp (D2) ) Let

{ x 1, . . . ,

x R} = Supp (D ) n Supp (D2) •

f^ (r e s p .

g^)

be a l o c a l e q u a tio n f o r

dimk c°x . ^ fi» S p ]

1=1

T h is makes sense because th e i d e a l w hich i s

its e lf.

some

•

(f^ ,

g^)

d e fin e s a subscheme o f

F

at

Supp (D 1) n Supp (D2) , i . e . ,

^ 4 .

N, and the dim ension i s f i n i t e . T h is i s

the in t e r s e c t io n number o f D1

and

D2,

and i t

is

check th a t i t

is

on cu rv e s , i t

depends o n ly on the d i v i s o r c la s s e s , n ot the d i v i s o r s :

P r o p o s it io n 1: (D ]

x^.

T h e r e fo r e o .

w hich has s t a lk

x 1, . . . ,

xn,

£x / ( f ,

and a t

x^

it

g)

at

is

isom orph ic to

V ( f i> gi> ' T h e r e fo r e , (D 1 • D2) = dim H °(F , ^

® o^)

=

® ° d 2)

=

- X[O p(-D .,) e C^,(-D2) ]

=

X( Op)

- X ( op ( — D i) )

+X (O p(-D .,- D g ))

“ X( op ( -D q) )

+

X( op ( —Di ~~ D g )) . QED

T h is m o tiv a te s : D e fin itio n :

Let (L.|

If

D1

and

L1

and

• Lg)

D2

L2

= X (O p )

be any i n v e r t i b l e - X ( L " 1)

a re any d iv is o r s on

- X (L ^ )

F,

i) ii) iii)

( (L ,

, )

+ X ( L ' 1 ® L g 1)

.

• c ^ (D g ))

.

i s a symmetric i n t e g r a l b i l i n e a r p a ir in g , i . e . ,

• Lg)

(L 1 ® Lj

F.

then

(D 1 • Dg) = ( ^ ( D , ) P r o p o s it io n 2 ;

sheaves on

= (Lg • L , ) • L g)

(L ^ 1 • L g)

= (L ,

= - (L ,

• Lg)

• Lg)

+ (L j .

• L g)

THE OVER ALL CLASSIFICATION OF CURVES ON SURFACES P r o o f:

(i)

i s o b vio u s, and ( i i i )

f o llo w s from ( i i )

85

i n v ir t u e o f

the ob viou s f a c t : (° p In fa c t ,

' L) = 0 .

I c la im : (Op(D)

f o r any curve

D

on

P. 0 -

• L)

= d e g p [L ®

Use the sequences: ^ ,(- D ) -

oF -

^

-

0

and 0 — L -1 ® Op(-D) -► L _1 — (L ® c ^ ) " 1 — 0

.

T h e r e fo r e , (Oj,(D)

• L)

= [x(O p ) = X(Op)

- X( Op( - D )) ] -

- X (L _1 ® Op( - ® )) ]

- X ( ( L ® ^ , ) ” 1)

= d egp tL ® c^ ] T h e r e fo r e , i f

[X (L -1 )

L 2 adm its a s e c t io n ,

.

( L 1 * L 2)

is

l i n e a r in

L 1,

by the

Riemann-Roch theorem (Theorem 1, L e c tu re 1 1 ). F in a lly ,

le t

i s any i n v e r t i b l e

£ (1 )

s h e a f on

S e r r e 's theorem s.

be a v e r y ample i n v e r t i b l e P , then

L (n )

sh e a f on

has a s e c t io n i f

n

is

P.

If

L

la r g e , by

Now by w r i t i n g the w hole th in g out one checks th a t the e x ­

p r e s s io n ( L 1 * L 2) + ( L ’ is

symmetric in the th re e v a r ia b le s

L 2 adm its a s e c t io n , i t o (n ),

i s a ls o

0

• L 2) L 1, when

- ( L 1 ® L» L.| and

• L 2)

L2.

Sin ce i t

LJadm its a s e c t io n .

is Takin g

0 when Lj

=

t h is im p lie s th a t ( L 1 • L 2)

But both

£ (n )

a re lin e a r in

and L2.

= ( L 1 (n )

• L 2) - (o (n )

• L 2)

.

L 1 (n ) adm its s e c t io n s , hence the two term s on the r i g h t T h e r e fo r e

( L 1 • L 2)

is

l i n e a r in

L2. QpD

T h is b i l i n e a r form on morphism on

P ic

(X )

fo r

X

P ic

(P )

ta k es the p la c e o f the d eg ree homo­

a cu rve.

I t in du ces th e f o llo w in g decom posi­

t io n : D e fin itio n : v e r tib le

P i c T (P )

sheaves

L

i s th e subgroup o f

(L a ll

• L r) = 0

L ' € P ic ( P ) .

D e fin itio n :

Num (P )

P ic

such th a t

= P ic

(F ) / P i c T (P )

.

(P )

c o n s is t in g

o f th ose i n ­

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

86

By d e f i n i t i o n ,

Num (F )

- the n u m erical d i v i s o r c la s s group o f

i s endowed w ith a n on -d egen erate symmetric i n t e g r a l p a ir in g in t o fundam ental r e s u lt co n cern in g p,

theorem

(fo r

F —

The

Num ( F ) , due t o S e v e r i and Neron, i s th a t i t

i s f i n i t e l y ge n e ra te d as an a b e lia n group; hence isom orph ic t o in t e g e r

z.

known as the base number o f

F.

Z p,

f o r some

We w i l l n o t need or p ro ve t h is

the b e s t p r o o f, how ever, c f . LANG-NERON, Am. J. Math. ,

1959,

R a t io n a l p o in ts o f a b e lia n v a r i e t i e s o v e r fu n c tio n f i e l d s )

2°

A lth ou gh t o understand th e w hole s it u a t io n con cern in g th e num eri­

c a l c h a ra c te rs o f a d i v i s o r c la s s

(D)

o r , e q u iv a le n t ly , a t the numbers

one must lo o k a t i t s

(o-p(D)

• L ),

fo r a l l

image in

L,

Num (F )

n o n e th e le s s f o r

most purposes some o f th ese numbers a re more im p ortan t and u s u a lly s u f f i c e :

o ( 1)

D e fin itio n :

If

r e la t iv e to

o (1 )

d eg (L )

i s a f i x e d v e r y ample i n v e r t i b l e

sh ea f on

F,

then

one d e fin e s : = (L • 0 (1 ))

and deg (D)

= d eg[ Op(D) ]

I n c id e n t a lly , i f F

=

D

d e g ^ c ^ ® o (1 ) ] .

is e ffe c tiv e ,

then

d eg (D) > 0:

le t

o (1 )

i(x ),

x

on

be induced by: i:

Let

H C

Pn

F

^

Pn .

be a hyper p lan e n o t c o n ta in in g any o f the p o in ts

g e n e r ic p o in t o f

Supp ( D ) .

Then th e curve

H1 = i * ( H )

dim (Supp (D) n Supp (H 1) )

=0

a

i s d e fin e d and •

T h e r e fo r e , deg (D) = (o p (D)

• O p d T ))

= (D • H ') > But suppose choose a i(y )

€ H

d eg (D) = 0 ;

c lo s e d p o in t y w h ile i ( x )

is

then

e Supp

0 •

Supp (D) n Supp (H 1) = #. (D)

and choose

s t i l l n o t in

H

T h is i s c e r t a i n l y p o s s ib le , and, t h e r e fo r e

f o r g e n e r ic p o in ts

H

such th a t

x € Supp (D) .

deg (D) > 0.

R etu rn in g to an a r b it r a r y i n v e r t i b l e

sh e a f

L

on

number o f g r e a t im portance i s i t s E u le r c h a r a c t e r is t ic . by an i n t e r s e c t io n p rod u ct t o o .

To p re v e n t t h i s ,

th e hyperplan e

F,

the o th er

T h is number i s g iv e n

To d e r iv e t h i s , use the t h ir d p a r t o f the

Riemann-Roch theorem on cu rves. P r o p o s it io n 3:

Let

n o n ic a l i n v e r t i b l e

L

be an i n v e r t i b l e

sh ea f on x (L )

F =

sh ea f on

F,

and l e t

g iv e n by Theorem 3, L e c tu re 11. i(L

• L ® sT 1) + x(Op,)

.

ft

be th e ca ­ Then

87

THE OVER-ALL CLASSIFICATION OF CURVES ON SURFACES P r o o f;

The form u la s t a t e s ; 2( x (L )

- X (O p ))

=

(L • L ® sT 1)

= - ( L _1

• L ® n_1)

= " x (°j? ) + X( L ) + x ( L _1 ® n)

- x (n )

or (# )

X (L )

If

- X (^ )

L -1 hag a s e c t io n ,

- x (n ® L ' 1) + x (n )

then

L = Op(-D)

0

0, —*■ D, ® L

=

o .

f o r sane curve

D.

Then

use the

e x a c t sequences; and

(c f.

Theorem 3, L e c tu re 1 1 ).

hence

(# )

fo llo w s whenever

F in a lly , l e t is if

n

is

By Theorem 2, L e c tu re 11,

be a v e r y ample i n v e r t i b l e F,

then

M~1(n ) and

la r g e , b y S e r r e 's theorem s.

the e x p re s s io n on th e l e f t i n (# ) [X (L ®

M) - x(O p)

+ x (^ )

= o,

L ” 1 has a s e c tio n .

£ ( 1)

any i n v e r t i b l e sh e a f on

—►0

o (n )

sh ea f on b o th

F.

have

If

M

s e c tio n s

Now a sim ple com putation shows th a t

i s lin e a r in

L.

Namely;

- x (n ® L " 1 ® M_1) + x (n ) ]

- [x (L )

- x(c^,) - x (n ® L _1) + x ( n ) ]

- tx(M )

- x(O p) - x (n ® M-1) + x (n ) ]

= +

(x(op) - x(L) - x(a ® L -1 ® M_ 1 ) Cx(Op) - x(M) - x(o ® L _1 ® M_ 1 )

-

fx (^ p ) - x ( L ® M) - x (n ® L -1 ® M_1) + x (n )

= =

But then the e x p r e s s io n in

(# )

th e f i r s t p a r t o f th e p r o o f.

( L -1

• n-1 ® L ® M) + (M-1

( L -1 ® M-1 0 is

• n -1

+ x(n ® M"1 ) + x(n ® L _ 1 ) ® L ® M)

• SI-1 ® L ® M)

.

0

fo r

T h e r e fo r e i t

L = M (-n ) is

0

fo r

and f o r

L = o (- n ) by

L = M. QED

T h is r e s u l t i s

the w eakest v e r s io n o f the Riemann-Roch theorem on F.

As one consequence o f t h is r e s u l t , we see th a t the o n ly r e a l l y im p ortan t n u m erica l c h a ra c te rs o f an i n v e r t i b l e

sh e a f

d eg (L )

= (L

L

a re

• 0 (1 ))

( L 2) = (L • L) and (L

• fl)

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

88

3 So far we have studied the discrete aspects of Pic (F), and hence the discrete aspects of the set of curves on F. To get at the exis­ tence questions of Lecture 2, we shall look at the continuous part of these two sets. The "glueing" which gives continuity must come from the concept of families of invertible sheaves and families of curves. We make the fol­ lowing definitions: Definition. Let S be a scheme (algebraic /k) . A family of curves on F, over S, is a relative effective Cartier divisor 3) C F x S, over S. A family of invertible sheaves on F, over S, is an invertible sheaf L on F x S: except that two invertible sheaves L 1, L2 will be said to define the same family of invertible sheaves if there is an invertible sheaf M on S such that: L, a L2 ® p2(M) . How does the concept of a family really provide the glueing? comes about because the collection of families forms a functor: a)

Curves F(S)

b)

Picp(S)

=

set of families of curves on

F

over

This

S

and

Given

=set of families of invertible

sheaves

onF over

S.

g

T —2— ► S, one obtains: P x T

►F x S ;

hence for 3) C Fx S (resp. L on F x S), (resp. h*(L) on F x T). This is a map g* a) Curve s^fS) --- ► Curves^fT) and b) Plcp (S) . PlOp(T) .

one obtains

h*( 2) )C Fx T

The glueing is now equivalent to the problem of representing these functors: to represent these functors is the same as to find a universal family of curves or invertible sheaves. And if you find such a family, say over S, then the set of k-rational points of S will be canonically isomorphic to the set of curves on F, or to the set Pic (F); i.e., you have put these sets together into whole schemes. Notice also that we have a morphism of functors: Curve Sp, — ► Plc-p which maps 3) C F x S to the invertible sheaf OpygC 3) ) • Consequently if C (resp. P) were schemes representing these two functors, one would auto­ matically get a morphism of schemes, cS —

p

which, on k-rational points, restricts to the obvious map from the set of curves on F to the set Pic (F) .

THE OVER-ALL CLASSIFICATION OF CURVES ON SURFACES

89

In terms of this glueing, we can say precisely why the numerical in­ variants of 1 ,2° are discrete. Say L1, L2 are two invertible sheaves on F x S. For each closed point s € S, they induce sheaves L« _ and Iy S L0 on the fibre F, and we can compute (L-1 ,_ 3•2L0 > 3 , 3_) : this number is constant on each connected component of S 1 [Since (L.I,S _ • L0 2,3_) is a sum of Euler characteristics and these are values of Hilbert polynomials, this follows from Corollary 3, Lecture 7.3 In other words, given any family of invertible sheaves over a connected base S, the image of each sheaf Ls in Num (F) is the same. Therefore, if an object P represents the functor PiCp, for each element of Num (F), the set ofinvertible sheaves inducing this element would form an open and closed set of P. The natural thing to do is to break up the functors PiCp and Curve^ accordingly into manage­ able pieces: Definition; Let | € Num (F) . For all schemes S, let PicJ,(S) be the sub­ set ofPiCp(S) consisting of those L on F x S such that for all closed points s e S, if L_ is the induced sheaf on F over s, then L_ has S » 3 numerical class |. Moreover, let CurveSp(S) be the subset of CurveSp(S) mapped by ® into PiCp(S) . Both form subfunctors denoted Curvesj, and Plc^, . The principal results at which we are aiming are: FIRST CONSTRUCTION THEOREM: For all t, Curves|, Is isomorphic to a functor h ^ ^ , where C(|) is a pro­ jective scheme. SECOND CONSTRUCTION THEOREM: For all Pic|, Is isomorphic to a functor h p ^ , where P(|) is a projective scheme. As a corollary, it follows readily that the full functors CurveSp and PiCp are represented by (non-algebraic) schemes which are the disjoint unions: II C( |) and H P( g) . I

5

LECTURE 13 LINEAR SYSTEMS AND EXAMPLES B e fo re lo o k in g a t the g e n e r a l problem o f c o n s tr u c tin g

C (£)

and

P ( g ) , we want t o d e s c r ib e some s p e c ia l ca ses in w hich the answer i s v e r y sim ple and

1 group

can now be t r e a t e d r ig o r o u s ly .

We s t a r t w ith a case in w hich the group

Num

Assume i )

H C F

i s an ir r e d u c ib le

ii)

F - H

is a ffin e ,

P r o p o s it io n 1: h

of

r (F

Then H;

- H, Op)

and hence the

P ic

(F )

P r o o f: t o nH

cu rve,

i s a unique f a c t o r i z a t i o n domain.

i s an i n f i n i t e c y c l i c group gen era ted by the

and P ic

le n t

P ic (F )

(F ) i s p a r t i c u l a r l y sim p le:

iii)

image

1 f a l l in t h is

then t o show how some o f th e Examples o f L e c tu re

c a te g o r y , hence

(F )

s Num ( F ) .

We must show th a t any d i v i s o r

f o r some i n t e g e r

n.

D

on

S in ce d iv is o r s

F

i s l i n e a r l y e q u iv a ­

a re W e il d i v i s o r s , e v e r y

d iv is o r is

the d if f e r e n c e o f two e f f e c t i v e d i v is o r s and we may as w e l l a s ­

sume th a t

D

is e ffe c t iv e .

L e t th e c lo s e d subscheme

D n ( F - H )

o f F - H

correspon d to the i d e a l 51 C R = r ( F - H, Op) Since 5i

in du ces a

p r in c ip a l i d e a l in each l o c a l i z a t i o n

fo llo w s th a t a l l prim e id e a ls a s s o c ia te d t o R

i s a UFD,

D

- ( f ) has

51

.

i t s e l f i s p r in c ip a l.

51

Let

n e it h e r z e ro e s nor p o le s in

51= ( f ) . F - H,

Rp o f

a re m in im a l ;

R,

hence,

it sin ce

Then the d i v i s o r

i.e .,

Supp [D - ( f ) ] C H.

T h is menas th a t D - (f) hence

D = nH.

T h e re fo re

rem ains t o check th a t

h

= nH,

g e n e ra te s

Num (F )

is

nH i s

v e r y ample f o r some

n e z ,

P ic

in fin ite

and th ese two groups a re iso m o rp h ic. sor

some

( F ) , and hence

But sin c e

n (i.e ., 91

Num (F ) .

c y c l i c —f o r then so i s Op(nH)

F

It

P ic (F )

is p r o je c tiv e ,

the d i v i ­

i s o f the form

o(1 )) .

92

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

T h e r e fo r e , as remarked in L e c tu re 12^ n(H • H) = (c ^ (H )

• Op(nH))

= (OpCH) • o ( l ) ) = deg H > and

t h e r e fo r e the image o f

h

in

0

Num (F )

has i n f i n i t e

o rd e r. QED

C le a r ly t h is r e s u l t a p p lie s t o r(

s in ce i f

P2,

P2 - H, o p ) g* k [X , Y ]

H

i s a h yp erp la n e,

.

2

T h e r e fo r e , a l l cu rves o

p (D) 2* £ p(d ) .

D

Sin ce

in

P2

in fh e homogeneous c o o rd in a te s curves on

have some d egree

H°( P2, o^ p ( d ) )

is

XQ, X 1^ X2

d , and

D = dH,

i.e .,

spanned by homogeneous form s

o f d egree

d, i t

fo llo w s th a t a l l

a re o f th e typ e we e x p e c t.

P2

I n c id e n t ly ,

th e P r o p o s it io n i s v a l i d in any dim ension, so i t

a p p lie d to v a r io u s Grassmannians, H yp erqu ad rics, e t c ,

can be

(a l s o t o h y p ersu rfa ces

o f some ty p e s , c f . ANDREOTTI, SAIMON, M on atsh efte fu r M a th .. 6 1 , 1957, p. 9 7 ). 2

o

a ls o f a i r l y

in cases where the P ic a r d group i s sim p le.

sim p le, the s e t o f cu rves i s

A c t u a lly , what i s alw ays sim ple are the f i b r e s in the

s e t o f cu rves over the P ic a r d group, i . e . , le n t to a f i x e d cu rve.

the s e t o f curves l i n e a r l y e q u iv a ­

However, t o s ta te t h e ir s tru c tu re p r o p e r ly , a g a in we

have t o fin d the glu e t o put th ese " l i n e a r system s" o f cu rves t o g e th e r . What i s r e q u ir e d i s fu n c to r

the f i b r e o f the morphism

$

from the fu n c to r CurveSp t o the

PiCp . Q u ite g e n e r a lly , G roth en d ieck has d e fin e d the f i b r e s o f a morphism

o f fu n c to r s . (S e t s ) .

ot e G(S) :

Let

Let

F, G F

G

be c o n tr a v a r ia n t fu n c to r s from a c a te g o r y be a morphism.

we s h a ll d e fin e the f i b r e

t o o , but n o t from o f o b je c t s o v e r

C

to (S e t s ).

S [i.e .,

of

Let

S

be an o b je c t in

over

a.

C,

t o the c a te g o r y ( S e t s ) .

]

C a ll i t

■£ 4a (T ----- ►S) = {0 € F (T ) | * ( p ) The r e s t o f the d e f i n i t i o n i s c le a r .

C/S

T — ►S, and a morphism

i s a commutative diagram

S

to and l e t

I t i s t o be a fu n c to r

I t i s a fu n c to r from the c a te g o r y

an o b je c t i s a morphism

C

= f* (a )

in

G (T )}

.

LINEAR SYSTEMS AND EXAMPLES In our ca s e , G.

over

Then

k

i s a g a in

because

p o in t i s

is

th is :

th e c a te g o r y o f a lg e b r a ic schemeso v e r

Spec (k ) If

P

a fu n c to r on th e c a te g o r y o f a lg e b r a ic

schemes The key

andG

th e f i n a l o b je c t in t h is c a te g o r y .

a re re p re s e n te d b y schemes

cc

X

q>: X -*■ Y ,

i s r e p re s e n te d by the a c tu a l

fib r e

cp” 1(a ) .

P iC p ,

th e f i b r e fu n c to r i s :

( P r o o f:

and

L

and

be an i n v e r t i b l e

sh ea f on

L in SysL ( S) = { ® C F x s | $ over

S

then

Q®

Y and

Let

a r e l a t i v e e f f e c t i v e C a r t ie r d i v i s o r

s) s

V ia th e u su a l maps,

P.

such th a t

P*(L) ® P^K)

some i n v e r t i b l e

sh e a f

K

for

on

S}

t h is i s a c o n tr a v a r ia n t fu n c to r in

. S.

In L e c tu re

1 we gave h e u r is t ic rea son s

fo r

a p r o j e c t i v e space.

The f u l l r e s u l t can now be

p roven :

P r o p o s it io n 2 ;

Y,

I s a c lo s e d p o in t o f

im m ed iate.)

In the case o f CurveSp, Let

and

is

*(M g)

t o s e c tio n s o f

c ) But as

Mg K( s)

=

( s ) , Mg

i n some n eighborhood o f

o^, th e s e c t io n 1

of

Opl i f t s

s. t o a s e c tio n :

a e r(U, P2j*(Ma)) = H°(P x U, M g) . d)

Then

a

d e fin e s a homomorphism: P; ( L S) - ^

in

F

x U. M oreover,

phism o f th e

s in c e a

induced sheaves

£

comes from La

and

* p. (L

fo r e

cp i s an isom orphism o f

cp

an isom orphism i n an open n eigh borh ood W

is

F x 3 —►S of

s

is

£

at

-1

p 2 ( s) .

I

T h e r e fo r e i f

s ' € Ug ,

L ,

L.

and

Lg

Then we have an open c o v e r in g

_* i

p0

( s) . T h eres,

S in ce

hence p2:

s

a re iso m o rp h ic. Sin ce

S

L a a re iso m o rp h ic. C a ll 3 ~ o f S such th a t p-j (L )

F x U^.

F i x isom orphism s p * (L )

P x

i s an isom or-

a l l p o in ts o v e r

of

S

a re isom orp h ic in each open s e t f)

in

£

con n ected , t h is im p lie s th a t a l l th e sheaves

t h is s h e a f and

and

cp

on th e f i b r e

F x Ug . e)

is

in Pp1( s ) ,

t o p o l o g i c a l l y c lo s e d , th e re i s an open neighborhood U C U * W 3 F x U . T h is p ro v e s th a t p . ( L „ ) and £ a re isom or-

such th a t

p h ic t o

)

1

£ H (s )

.

Then in

P x (U^ n U j ) ,

^ °

£ i s an automorphism o f

p * (L ).

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

96

T h is i s g iv e n by m u lt ip lic a t io n by a u n it : ° ij

€ r (p x % M

n

r(U± n uj( o^) (c f.

L e c tu re 1 1 , T V ).

in g

{U^}.

sh ea f

K

Then

i s a 1-C z e c h -c o -c y c le on

S

f o r the c o v e r ­

L e t t h is c o - c y c le be the t r a n s it io n fu n c tio n s f o r an i n v e r t i b l e on

S.

Then i t

p h ic g l o b a l l y t o

fo llo w s from our c o n s tr u c tio n th a t

£

i s isom or­

p * (L ) p^ (K ) . QED

T h is r e s u lt i s c l o s e l y r e la t e d t o the see-saw p r i n c ip le o f LANG (c f.

h is A b e lia n V a r i e t i e s ) .

C o r o lla r y :

If

H1(F , o^)

union o f a ( i n f i n i t e ) CurveSp

= (0 ),

then

P ic ^

i s re p re s e n te d by the d i s j o i n t

d is c r e t e s e t o f p o in t s , i . e . ,

o f Spec (k )> s .

i s re p re s e n te d by th e d i s j o i n t union o f p r o j e c t i v e

T h e re fo re

spaces ( o f v a r i ­

ous dim ensions) . T h is com pletes our j u s t i f i c a t i o n o f our d e s c r ip t io n o f cu rves on P2 .

Perhaps t o add th e l a s t p o in t , we should compute: ( o (n )

[im m ediate b y b i l i n e a r i t y ,

E x e r c is e :

= n • m

.

and th e check: ( 0( 1 )

f o r two d i s t i n c t lin e s

* o(m))

E , , H2

• 0( 1 ) )

in

- H2) = 1

= (E,

p2 ] .

W r ite down e x p l i c i t l y the u n iv e r s a l f a m ilie s o f cu rves on

P2 .

W ith ou t p r o o fs , we want t o supplement Examples 2 and 5

F u rth er Exam ples:

o f L e c tu re 1 b y r e l a t i n g th e r e s u lt s th e re t o our p re s e n t th e o ry . th ese s u rfa c e s a re " b i r a t i o n a l ” t o open dense su b sets.

In f a c t ,

it

i.e .,

P2 ,

a re isom orph ic t o

Both o f p2

on

fo llo w s from t h is th a t

H1 (F , Op) = H ^ F , Op) = (0 ) in b o th th ese c a ses. Now, i n th e case

T h e re fo re b o th f a l l under

F =P 1 x P 1 ,

th e C o r o lla r y ju s t g iv e n .

then

P ic (F ) = Num (F )

^

Z ®z

.

I n f a c t , a b a s is i s g iv e n by th e two sheaves L1 = and th e d eg rees d

and

e

d

and

e

and L2 = P ^ * 1) ) o f a d iv is o r

D

d e s c r ib e d b e fo r e a re ju s t the

d e fin e d by: Op(D) as L® ® Lg •

LINEAR SYSTEMS AND EXAMPLES The p a ir in g i s

g iv e n by

(L,

•L,) = 0

(L i

• L 2) = 1

(Lg •L2)= 0 . Now i n case where

F

is

o b ta in e d by

P ic I n f a c t , a b a s is i s

(F )

b lo w in g up two p o in ts in

s Num (F )

s Z ® Z ® Z .

g iv e n b y th e th r e e sheaves L = SpW

M2 =

•

The p a ir in g i s g iv e n b y: (M,

•M ,)

= -1

(M n

•M2 )

=

0

(M.,

L)

= 1

(M2

•M ,)

= 0

(Mg

•M2 )

= -1

(Mg

L)

= 1

(L

•M ,)

=

(L

•Mg)

=

(L

L)

= -1

1

1

LECTURE 14 SOME VANISHING THEOREMS Some o f the d e e p e s t r e s u lt s i n a lg e b r a ic geom etry concern the p ro b ­ lem o f g i v i n g c r i t e r i a f o r th e h ig h e r cohom ology groups o f a sh e a f t o be

0.

The p i v o t a l r o l e p la y e d b y th ese r e s u lt s i s due t o th e f a c t th a t the E u le r c h a r a c t e r is t ic o f a c o h eren t s h e a f on some v a r i e t y i s g e n e r a lly v e r y comput­ a b le : e i t h e r d i r e c t l y ,

o r b y use o f the v e r y p o w e rfu l H irzeb ru ch -G ro th en d ieck

form o f th e Riemann-Roch Theorem;

on the o th e r hand, i t

i s u s u a lly the group

o f s e c tio n s o f such sheaves w hich has g e o m e tric i n t e r e s t and d i r e c t s i g n i f i ­ cance.

T h e r e fo r e , whenever one can p ro ve th a t the h ig h e r cohom ology i s

o,

one should e x p e c t many c o r o l l a r i e s . A f i r s t theorem o f t h is typ e was p ro ven in L e c tu re 11 .

The g e n e r a l

problem was fo rm u la te d by th e I t a l i a n s : i t was known as the problem o f p o stu ­ la t io n

(i.e .,

when does th e dim ension o f som ething tu rn out t o eq u a l th e num­

b e r which one had p o s tu la te d .1? ) .

P ic a r d p roved b y a n a ly t ic methods a v e r y

famous r e s u l t o f t h is kind (th e theorem o f the r e g u l a r i t y o f th e a d jo in t , c f ZARISKT's book on s u r fa c e s ); t h is r e s u l t was g r e a t l y exten ded by KODAIRA in one o f h is most famous papers (P r o c . N a t l. Acad. S c i . ,

1953, p.

1268: A

d i f f e r e n t i a l - g e o m e t r i c method in th e th e o r y o f a n a ly t ic s ta c k s ) , and tod a y it

i s known as K o d a ir a 's v a n is h in g theorem .

A n oth er r e s u l t in t h is d ir e c t io n

i s S e r r e 's d u a lit y theorem ( v a s t l y exten ded by G r o th e n d ie c k ): d i r e c t descen den t o f R o c h 's r e s u l t and i t s in g u la r v a r i e t y , how t o compute an

H1

te lls ,

t h is i s the

on an n -d im en sio n a l non-

b y means o f an

Hn~i ,

w hich a t

l e a s t cu ts the problem in h a l f . We s h a ll p ro ve h ere (w it h th e h e lp o f tech n iq u es d eve lo p e d and used by N ak ai, Matsusaka and K leim an)

o n ly a weak v a n is h in g theorem , bu t one w hich

i s u n ifo rm ly a p p lic a b le t o a la r g e c la s s o f sh eaves. s h e a f on

Let

?

be a coh eren t

pn :

D e fin it io n :

5

i s m -re g u la r i f

Hi ( Pn , ^ ( m - i ) )

= ( 0) fo r a l l

i > 0.

T h is a p p a re n tly s i l l y d e f i n i t i o n r e v e a ls i t s e l f as f o llo w s : P r o p o s it io n :

(C a s te ln u o v o )

Let

?

be an m -re g u la r co h eren t sh ea f on

Then a)

H °( Pn , ? ( k ) )

is

spanned by

H °( Pn , 5 ( k - i ) )

® H °( Pn , 0 ( 1 ) ) 99

if

k > mj

Pn .

100

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE b)

Hence

P , 5 ( k ))

a 1)

y (k )

= (0 )

whenever

i s g en era ted as

P r o o f;

k + i > m .

£ p -module by i t s

k > m.

g lo b a l s e c tio n s i f

n

We use in d u c tio n on

In g e n e r a l, g iv e n

i > 0,

5,

p o in ts i n th e f i n i t e

n:

fo r

choose a hyperplan e set

A( y ).

H

n = 0, the r e s u lt i s

o b vio u s.

n o t c o n ta in in g any o f the

Tensor the e x a c t sequence:

O - ^ o p (-H ) ■ ♦ O p n n

£ jt

0

21! £ w it h

y (k ).

For a l l

x € Pn ,

then m u lt ip lic a t io n b y

f

p

if

n

M ) f

i s a l o c a l e q u a tio n f o r

i s i n j e c t i v e in

i s a u n it a t a l l a s s o c ia te d prim es o f (* )k

o - * y (k - i) - * ( k ) -

i s e x a c t.

T h is im p lie s th a t i f J jj.

( ? oH) ( k ) -

H

x, f

T h e re fo re the r e s u l t i n g sequence o

y

-►

y H( m - i ) )

Hi+1 ( 5 ( m - i - 1 ) )

i s m -r e g u la r , the sh ea f

3^

on

H

H £* Pn__1, we use the in d u c tio n h y p o th e s is t o o b ta in

. i s m -re g u la r. a) and

b) f o r

In p a r t ic u la r , use HL+1 ( 5 ( m - i- 1 )) - H 1+1( 5 ( m - i ) ) — H^+1 ( ? H( m - i ) )

If

at

s in ce by c o n s tr u c tio n ,

In p a r t ic u la r , we g e t ;

y(m - i)) Sin ce

?x .

3^.

i > 0,

byb)

r e g u la r .

fo r

i s ( 0) .

f i r s t group

3 ^ , th e l a s t group i s

C on tin u in g i n t h is way we p ro ve To g e t

( 0)5

b y m - r e g u la r it y the ( 0 ) and

T h e r e fo r e , the m id d le group i s

.

3

is

(m + i)-

b) fo r y.

a ) , lo o k a t the diagram : H °( J (k - 1 ) )

( 1))

® H °(o

-2— H °( Sw( k - 1 ) )

® H ° (o h( i ) )

n t

\l

H °( ^ ( k - l ) ) Note th a t t

is

-

H °(

a

is

5 ( k ) ) -----------------------------► H °( f H( k ) )

s u r je c t iv e i f

s u r je c t iv e i f

i s the whole o f H °( y ( k - l ) ) .

k > m

H °( 3H( k ) ) ,

k > m

because

H1( S ( k - 2 ) )

by c o n c lu s io n a) f o r i.e .,

H °( y ( k ) )

is

3^.

= (0 ).

T h e r e fo r e ,

spanned by

Im ( m.)

But l e t

Then the Image o f H °( 3 ( k - i ) ) . s u r je c t iv e and

M oreover v (Im \i)

h € H °( Pn , £ p ( 1 ) ) be th e g lo b a l eq u a tio n o f n H^( 3 ( k - i ) ) in H ( 3 ( k ) ) i s morep r e c i s e l y h

I n o th e r w ords, t h is i s p a r t a) i s p roven f o r

y .

of

Im n

to o . T h e re fo re

and by H.

\± i s

101

SOME VANISHING THEOREMS Now by S e r r e 's theorem , we know th a t s e c tio n s p ro v id e d th a t im p lie s th a t modules i f

o

and w it h

la r g e enough.

? (k )

i s g en era ted b y i t s

P u ttin g t h is to g e th e r w ith a) of o p n o p (1 )

x:

t h is i d e n t i f i e s o p (k-m) w ith o_ at x, and ? (k ) o n n x. Then H (o_ (k -m )) becomes ju s t a v e c t o r space o f e l e -

at

ments o f th e l o c a l r i n g £x

is

H°( ^ (m )) ® H °(o p (k -m )) g e n e ra te s the sh ea f 5 (k ) “ n k » o. But f o r e v e r y x e Pn , f i x an isomorphism o f

at n 5(m)

k

n and the statem ent sim p ly says th a t

o ,

—x '

g e n e ra te s th e s t a lk

5r(m )x ,

i.e .,

5(m)

o H ( ^ (m ))

i s g e n era ted b y i t s

g lo b a l

s e c tio n s . QED Our main r e s u l t i s : THEOREM:

For a l l

n,

th e re i s a p o ly n o m ia l

Fn ( x Q, . . . ,

such th a t f o r a l l coh eren t sheaves o f id e a ls if

aQ, a 1, . . . ,

an

4

on

x n)

PR,

a re d e fin e d b y: n

x(J( m)) = £

ai ( i )

,

i= o then

4

P r o o f: is

o b vio u s.

is

H

aR) - r e g u l a r .

A g a in we use in d u c tio n on

G iven

a h yp erplan e

Fn (a Q, a 1, . . . ,

$,

le t

such th a t

Z C PR H

n

i s d i s j o i n t from

th e e x a c t sequ en ce: ®h

( *)m

*(m)---------- ► j (m+i ) -*■ ( j ® oH)(m+i)

0

sin c e f o r

n = 0

th e r e s u lt

be the c o rresp o n d in g subscheme; A (o z ) .

choose

As ab ove, we g e t

0

% w hich i s i n j e c t i v e H

on the l e f t

i s i n j e c t i v e i n the s h ea f

hand,

4„

s in c e m u lt ip lic a t io n by a l o c a l e q u a tio n f o r

4,

as i t

i s a s h ea f o f i d e a ls on

7 rL eq u a tio n f o r

H at

x.

H:

i s a subsheaf o f le t

x e P^ n

o p .

and l e t nf

On the o th e r be a l o c a l

Then 0

jx

- x , Pn

—x ,Z

0

g iv e s : T o r 1( - / f ‘ - x » —x,Z^ ■*

- x ,H

by t e n s o r in g w ith

o x / f ' o x = ox H.

s in c e

o „ 7 (s in c e f i s a u n it a t a l l a s s o c ia te d Xy T h is shows th a t i s a sh ea f o f id e a l s , and we can use

f

prim es o f in d u c tio n .

i s not a o - d iv is o r in ox z ) .

And

T o r 1 (o x / f- o x , ox z )

= (o )

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE Now, by ( * ) m, X (3 H(m +1))

= X ( 3(m+1) )

=

I

- X (3 (m ))

ai [ ( “ i 1 >

■ ?

" ( i )]

, , , ,

.

i=0 T h e re fo re we can assume th a t a b le p o ly n o m ia l

G-

^

is

GKa.,, a 2, . . . ,

depending o n ly on

n.

Put

an) - r e g u l a r , f o r a s u i t ­

m1 =

an) .

Then we

g e t , b y ( * ) m:

(i)

0 -

H°( 3(m)) - H°( 3(m+1) )

H°(3H(m+l ) )

— H1( 3(m)) — H1(3 (m+1) ) fo r (ii)

m > m.j-2. And f o r any

0

H^( 4(m )) -* ^ ( ^ m + l ) )

fo r Now s in c e

= (0 ),

H ^ (4 (m ))

th a t as f a r as

fo r

= (0 )

If

m >

m1-2 ,

i > 1

th a t

fo r

and

m»

0, t h is .la s t sequence ( i i )

i > 2

a re concerned,

and 4

then e it h e r

m = m2, where

pm+1

is

T h is means

m1- r e g u la r .

On

s u r je c t iv e or

dim H1 (0 (m )) m2 >

,

pm

P r o p o s it io n we know th a t

. is

s u r je c tiv e .

By the

2

H °(3 H(m2) )

i s su rjective.

m > n ^ - i.

i s a ls o

t e l l s us:

dim H1 (4 (m+1) ) < But suppose

we g e t :

-►0

as soon as

TP

H2, H3, . . . ,

the o th e r hand, sequence ( i ) (# )

i > 2,

m > m1- i .

H ^ K m ))

t e l l s us th a t

—0

® H ° (^ j(i)) -

H °(3 H(m2+ 1 ))

Therefore i t follow s that the image of

H°(3(mp) ) ® H °(op (1 )) / n

in

H °( $ (m2+ 1 ))

tio r i, Pm

is

(# ')

pw is m2+1

i s mapped s u r j e c t i v e l y onto s u r je c t iv e .

s u r je c tiv e , i t If

is

m > ml - 1,

as a fu n c tio n o f

H°( 4H(m2+ 1 ) ) .

In o th e r w ords, lo o k in g a t a l l

s u r je c t iv e f o r a l l la r g e r dim H1( 4 (m )) i s m,

m.

Hence, a f o r m > mn, i

once

Hence:

s t r i c t l y d e c r e a s in g ,

u n t i l i t rea ch es

0.

T h e re fo re c l e a r l y : 4

is

[m ^ dim H1( J (m1- i ) ) ] - r e g u l a r

Up to t h is p o in t , we have n ot used the f a c t th a t i d e a ls .

But now we compute:

.

$

i s a sh ea f o f

SOME VANISHING THEOREMS

103

= dim H°( I (m1- 1 ) )

dim H1( d (m1—1 ))

< dim H °(o p (m1 - 1 )) = H (a0 , a 1, . . . , where

H

i s a p o ly n o m ia l i n th e

a 's

and in

G K a ^ . . . , an ) + H (aQ, . . . ,

an ;

m1 .

^ (n ^ - 1 ) )

- x(

- x( )

In s h o rt,

an j G ( a , , . . . ,

j(m 1- 1 ))

4

is

an ) )

r e g u la r . QED A few rem arks:

F ir s t o f a l l ,

assumed t o be a sh e a f o f id e a l s .

the theorem i s f a l s e u n less

Thus, take

n = 1,

= £ P 1( + k) 0 £ pi ( “ k) Then

X( y k (m ))

such th a t

5^.

= 2 (m + l), w hich i s m -re g u la r i s

Second, suppose we a re j e c t i v e a lg e b r a ic scheme F ix

an im m ersion

i s in depen den t o f k:

is

hence

X C Pn ,

and l e t

M oreover, l e t

K

and say x^) x(

r = dim X;

then th e re

$ C ov

such th a t i f

0(m ))

j?

= £j__0

p ro­

tru e —

rn &j_( ^ ) ,

is then

ap ) - r e g u l a r .

f o r a g iv e n 4,

Z C Pn ,

m

concerned w ith the geom etry on a f i x e d

F (x n, . . . ,

F (a Q, . . . ,

To p ro ve t h i s ,

bu t the l e a s t

m = |k| - 1 .

any sh e a f o f i d e a l s , and 4

is

•

X; then the analogous r e s u lt i s

i s a p o ly n o m ia l

4

and l e t

le t

i

d e fin e the c lo s e d

5 be the sh e a f o f id e a ls on

be th e sh ea f o f i d e a ls on

PR

Pn

d e fin in g

subscheme 7 C X, d e fin in g Z. X.

Then one has

the sequence:

I t f o llo w s th a t i f mQ+ 1,

then

$

5

is

is

mQ- r e g u la r , and

mQ- r e g u la r

H ^ K (m ))

as a sh ea f on

x ( 5 (m )) -= x (

X. But

= (0 ),

fo r

i+m =

sin ce

$(m )) + x ( K (m )) independent o f 4

the c o r o l l a r y fo llo w s from th e theorem . t io n th a t k > 0,

H °(

and th a t

I t a ls o fo llo w s from th e P r o p o s i­

J (mQ+ k ) ) 0 H °(o x ( i ) ) -»■ H°( § (mQ+ k + 1 )) i s s u r je c t iv e i f s (m)

is

ge n e ra te d b y i t s

g lo b a l s e c tio n s i f

m > mQ.

LECTURE 15 UNIVERSAL FAMILIES OF CURVES o f L e c tu re 12

We a re now rea d y t o p ro ve th a t the scheme

C (| )

e x is ts .

F ix a n o n -s in g u la r p r o j e c t i v e

and f i x an embedding

F C Pn .

As u su a l, l e t

In L e c tu re

o( 1)

S

F,

be the induced v e r y ample i n v e r t i b l e

sh e a f.

12, we made the d eco m p o sitio n : C u rv e ts )

(fo r

s u rfa ce

connected ) .

=

H |eNum(F)

Curve s| (S )

A c t u a lly , f o r the purposes o f t h is p a r t ic u la r p r o o f

we w i l l o n ly need a c o a r s e r d e co m p o sitio n .

In f a c t ,

g iv e n

DCF,

we w i l l

o n ly lo o k a t the H i lb e r t p o ly n o m ia l: P (n )

= xCOpC-D + n ) )

In v i r t u e o f P r o p o s it io n 3, L e c tu re 1 2 , image and

§ b)

(# )

of

D.

In f a c t ,

the a r ith m e tic genus o

P (n )

P (n )

.

i s determ in ed by the n u m erical

i s determ in ed by a)

Pa ( D ) •

T h is i s

9_f ( - D + n) — c ^ (n ) - c ^ (n ) — o

the d eg ree

d

of

D,

seen by ,

hence

P(n) = x(c^,(n)) - x(c^(n)) = x (O p (n ))

- d • n - 1 +Pa ( D ) .

In any c a se, we w i l l use the d eco m p o sitio n :

IIp

Curves-^ S) = --------- F (fo r

S

OpxS(-D )

c o n n e c te d ), where

P CurveSp(S )

has H ilb e r t p o ly n o m ia l

i s n o t con n ected , then say

P

S =

is

the s e t o f

on each f i b r e . S^,

C u rves?( r S)

C u rves£(S) F

=

II a

where

Sa

CurveSp(S )

D C F x S

such th a t

To be p r e c is e , i f

.

I t i s v e r y ea sy to check th a t t h is i s a su b fu n ctor o f CurveSpj and i f i s re p re s e n te d by a (a lg e b r a ic )

scheme

105

C (P ),

then

S

i s con n ected , and l e t

C (P)

t h is

i s a d i s j o i n t union

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE of open subsets C(|) representing the various sub-functors CurveSp . Now fix some P. (I.) By Lecture ik, '\b9 there is an m0 mQ depending only on P, such th at i f D C F is any curve giving the H ilbert poly­ nomial P, then Op(-D) is m0-reg ular. We may as w ell also assume th at H1(Op(m0)) (o) •. H1(°p(m0) ) = (0) Then we conclude: (a) H1 H^Opt-D (Op( -D + mQ)) m0)) = H^Opf-D H^OpC-D + mQ)) m0)) = (0), and Op(-D + mQ) is spanned by i t s sections. Using the exact sequence (#) for n = mQ, we also conclude: (b) H1 (^(m 0)) = (o). (II.) Now suppose D C F x S is any family of curves giving the H ilbert polynomial P. F irs t of a ll, we get: (h)s P*(O£)(m0) ) is lo cally fre e , of rank (b)s r = x(Op(m0) ) - P(m0) P(mQ) , (depending only on P ), and the formation of p* commutes g with base extensions extensions TT ---------- SS . This follows from (b ), from Corollary 11,, 3°, Lecture 7, 7 , and from the exact sequence (#). The useful consequences of (a) w ill be: (a)g

R

+

=

>

and

P*P*[O fxs ("D ++ mo^ mo^]] °FxS( °FxS(“D “D ++ m0^ isis surj surjective ective •• P*P*[°FXS("D The ffirirsstt is trueis by Corollary true by C1 ,orollary 3°, Lecture 1, 3°, 7;Lecture and 7;the second and the is second true is true because P*t°pxs(~D P*[°jrxS^~D + mo ^ maps onto H°(Op(-Dg+ mQ)) for a l l closed points s € S;S; and and H°(Op(-Dg H°(Op(-Dg ++ mQ)) mQ)) generates generates opop(-Dg (-Dg+ +mQ)mQ)= = S(-D S(-D + mQ) mQ)00 K -® K(( 3s ).) . -® (III.) Again suppose D C F x S is a family of curves. From the sequence (#) for n = mQ and (a)g> we ge t: ■4-

n a n n o M rtQ

(M \

m

yn

o

( a\

t.tq

+■ •

a

211

“S 1 Fixing a basis eQ, e ^ . . . , of H°(Op(m0) ) , we have determined: a) a lo cally free sheaf P*[Op(m0)] of rank r , b) N + i-sectio n s s^ = a(i ® e^) which span P*[^,(m0)].

UNIVERSAL VAMILIES OF CURVES

107

T h is i s an S -va lu ed p o in t o f the Grassmannian (h ) g ,

th e fo rm a tio n o f

P*[O p(ia0) ]

G^

In v ir t u e o f

i s f u n c t o r i a l in

S, and the

w hole proced u re d e fin e s a morphism o f fu n c to r s :

P

o

Curve s-m ----- ►h^ ^F % r (IV .) of r

f S ----- ►G^ r

Now suppose we a re g iv e n an S -va lu ed p o in t ,

G^ r and*

.

Then

f

d e fin e s a l o c a l l y f r e e

(N + 1 ) - s e c t io n s

sQ, . . . ,

S^

s h ea f

£

&.

spanning

o f rank

T h is d e fin e s

a s u r je c t iv e homomorphism: O g g fl^ O p O O ,,)) Let

K

be the k e r n e l o f

—

a.

* ^

*0 *

Then p u llin g up v i a

p:

F x S -*■ S,

we o b ta in P*( X > -

P * [ o s j» H °(^ ,(m 0) ) ]

-

p* 6

-

0

^FxS^“V D e fin e

3

id e a ls on

to

be th e image o f

F x S.

p * ( K ) ( - m0)

in Op^g:

a sh ea f o f

T h is w hole procedu re d e fin e s a morphism o f

fu n c to r s : tu,

— ---- ► A l l Sub scheme Stp

\

(V .)

What i s

as in ( I V . ) .

Y 0 o ? Then

r

----------------------------- P

S t a r t w ith

D C F x S,

and c o n s tru c t

f

f o llo w in g the p roced u re o f ( I V . ) :

and K —

+ mo ^

But we saw i n ( a ) g th a t th e sub sh e a f was spanned b y th e s e c tio n s in t h is K, in

^px g ( m0)

O p^gt-D );

i s e x a c t ly

0£yg(-D + mQ) .

i.e .,

T h e r e fo r e

i

is

i.e ., Y o

n a tu r a l in c lu s io n o f Curves in A l l

(V T .)

+ m0) o f —FxS^m0^ the image o f p * ( K )

Sub schemes

We can a b s t r a c t th e r e s t o f th e argument:

g iv e n th e set-u p

o f morphisms o f fu n c to r s (fro m the c a te g o r y o f a lg e b r a ic schemes /k

t o th e c a te g o r y o f s e t s ) ,

assume th a t:

108 (#)

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

a e

fo r a l l a ll T

B (S), th e re i s a subscheme *■ S,

( ( \

g*(cif) € B(T) i s in th e su b se t

A( T)

/

Then th e re i s a subscheme th e in c lu s io n of

h^

in

Y CS

\ / g f a c to r s I I through Y

V

GQ C G such th a t

such t h a t f o r \

/

I

A s L

h^ .

0

,

$ b e in g

(P ro o f l e f t to th e r e a d e r .) (V II.) (# ) q

We must v e r i f y (#) .

In our ca se , i t means

f o r a l l c lo se d subschemes Z C P x S, such t h a t f o r a l l T ——►S,

j

Z x TCF x T

y

fam ily o f curves over T, whose sh eaf o f i d e a ls has H ilb e r t polynom ial P

is a

th e re i s a subscheme

Y CS

\

j

I g f a c to r s \ I through Y j

But by th e key r e s u l t on f l a t t e n i n g s t r a t i f i c a t i o n s , th e re i s a JL C S o O U . U 1 1 toh i j ja c l ot d x x j. ± u v c r x , w iith o n H x u ilb x u cerr t u sub scheme Y such Z T i sa -fl jl- cai ot over T, w a polynom ial X(oZxT )) "- pP(n) X(-ZxT (n )) = x (o (£ p (n )> (n ) S i f and only i f g f a c t o r s th ro u g h Y. I t rem ains to analyze when X x T i s a c tu a lly a C a r tie r d iv is o r . This i s d e a lt w ith by: S Lemma: L et Z C F x T be a cclo lose sedd subscheme, subscheme, ffllaa tt over over T. T. LLet et t t € € TT be be a clo se d p o in t such t h a t Z^ i iss aa curve curve on on F. F. Then Then ththeerere i iss an an open open nneeigighh­­ borhood U of t in T such t h a t Z n (F x U) i s a C a r tie r d iv is o r on U. P ro o f: Since p: F x T -► T i s a c lo sed map, i t s u f f ic e s to prove onen neighborhood ffi of F x ( t) i n which Z i s a C a rtie r tie rr tth h a t th e re i s an open d.Jiv oXr .. L etU -ft. x fce I*F xX T be any inl l Ut asuch a t p(x) = tO.. L etU 3$x LC £x be U .V J is LBU ± UC C L U y jp J Uo- L u u i l th 0 1 J£LO = ±jfc? £ x u c th e i d e a l d e f in in g Z a t x , and l e t m^ C be th e maximal id e a l. Since ox /mt • ox i s th e lo c a l r in g o f x on F x ((tt)),, and and sin sincce e Z^ Z^ iiss aa C ar­ t i e r d iv is o r , Jx + mt • °x °X ■ r

z

the r e s t r i c t i o n

to f

S

of

such th a t f

to

sure o f the graph o f

CQ.

f Q,

fQ

such th a t:

R/*>

in

i t s q u o tie n t f i e l d ,

in du ces a morphism

m • (R f p ) ,

y = f(z ). Then

fQ

Let

f(x )

then

p

d e fin e s a

C0 = f _1 ( Y ) , and l e t f Q be

i s a CQ-v a lu e d p o in t o f Y,

Y which

i . e . , because in the c l o ­

4 Y.

We s h a ll show th a t t h is i s absurd. T h e r e fo r e

n U .

►U C Y .

prim e i d e a l l y i n g o v e r

e C

y,

dim[R /* • Rp ] = 1

i s n o t the r e s t r i c t i o n o f a C -valu ed p o in t o f

Curves^ .

.

C i s a 1 -d im en sio n a l n o n -s in g u la r v a r i e t y ,

Then

and th e g iv e n homomorphism from

If

GN

F ix

some prim e i d e a l p

be th e i n t e g r a l c lo s u r e o f the domain

and l e t

as a su bset o f

31 C m d e f in in g the c lo s e d subset (Y - Y)

an i d e a l

p C m,

Let

Y

y € Y - Y.

U = Spec ( R ) , an a f f i n e neighborhood o f y € Y ,

ii)

Then i t

Let

Then p ic k a c lo s e d p o in t

But

h^

i s isom orp h ic to

d e fin e s a fa m ily o f cu rves Dq C F x C0

( g i v i n g the p o ly n o m ia l over

C.

But s in c e

and d i v i s o r s on

p) w hich i s n o t th e r e s t r i c t i o n o f a fa m ily o f cu rves

C and

F

a re n o n -s in g u la r ,

F x C

F x C a re th e same as W e il d iv i s o r s .

Dn , as a W e il d i v i s o r , be w r it t e n out a s:

i s n o n -s in g u la r,

In p a r t ic u la r ,

le t

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

Bo - I ni Zi , 0 where

Q

i s a c lo s e d subset o f

th e c lo s u re o f

Z^ Q

in

an e f f e c t i v e d i v i s o r on because i t s T h e r e fo r e C to

F x C. F x C.

F x CQ Let

Then

D

D

Let

be

is c e r ta in ly

M oreover, i t i s a r e l a t i v e d i v i s o r o ver

support does n o t c o n ta in any o f the f i b r e s i s a fa m ily o f cu rves o v e r

i s con n ected , the H ilb e r t p o ly n o m ia l o f P.

o f codim ension 1 .

D = E

C

e x te n d in g Op( -D)

T h is c o n t r a d ic t io n p ro v e s th e theorem .

is

F x iz), DQ.

C

z e C.

F in a lly ,

sin ce

c o n s ta n t, hence e q u a l

LECTURE 16 THE METHOD OF CHOW SCHEMES Sin ce the e x is te n c e o f th ese u n iv e r s a l f a m ilie s has such p i v o t a l im portance in the p r o o f o f the main e x is te n c e theorem s, i t

seems rea so n a b le

t o sk etch th e o n ly o th e r known approach t o t h e ir c o n s tru c tio n —th a t o f Chow and van d er Waerden. fix

the d eg ree

d

A ga in l e t

o f a curve

F C Pn DCF,

be g iv e n .

In t h is approach we o n ly

n ot the p o ly n o m ia l

p

as above, i . e . ,

we decompose: Curve s^CS) p (fo r

S

c o n n e c te d ), where

such th a t th e induced cu rves Say

X

II Curve s^ (S ) d> o F

CurveSp( S) Dg

is a p r o je c tiv e

D iv x (S )

=

stands f o r the s e t o f

on th e f i b r e s a l l have d eg ree scheme:

= ( 3) C X x S|

D a r e l a t i v e e f f e c t i v e C a r t ie r ^

g e n e r a liz in g CurveSp .

In some cases where

t o study than

f o r some s u rfa c e s

Curves^

d.

then we can d e fin e a fu n c to r :

d iv is o r over

Grassmannian

D C F x S

S

J

dim (X ) > 2, t h is may be e a s ie r F.

For exam ple, i f

X

is a

G, the methods o f L e c tu re 13 en ab le one to prove th a t D iv n

where

D

broken up

i s a d i s j o i n t union o f p r o j e c t i v e

Ir

in t o D iv ^ ,

spaces.

In

k > 0,

and

one f o r each in t e g e r

f a c t , D iv n i s 1/^ D iv^ i s ju s t a

l in e a r system . The method o f Chow i s t o c o n s tru c t a morphism o f fu n c to r s : d Curve Sp f o r th e Grassmannian

G = Gn n_i .

®

d ► DiVg

To do t h i s , we f i r s t c o n s tru c t a subscheme

Z C pn x Gn ^n_ 1 H e u r i s t i c a l l y , e v e r y c lo s e d p o in t o f space

L C Pn

o f dim ension

n -2 .

Gn n-1

correspon ds t o a lin e a r sub­

P u t t in g th ese to g e t h e r , th e y form

To be p r e c is e , r e c a l l from L e c tu re 5 th a t 111

GR n-1 = P r o j

(R ),

where

Z. R

is

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE a graded r in g g en era ted by elem ents pi If

j < k

i

...

i

’

9 n-1

o < 1

< i

< ... < 1

< n .

a re the two in t e g e r s om itted in th e sequences o f

i 's ,

we can

s im p lify n o ta tio n by p u ttin g

k = ^ -Lii f-L2 i 9' '

J Then

Z

*9 i n-1

i s d e fin e d as the scheme o f z e ro e s o f the s e c tio n s k-1 3k =

n

{ I

of

p ^ (o (i))

of

Pn_ 2,s

j=k +i

® P 2( o ( i ) ) ,

Z

fo r

Gn n -1 9

over

C la s s i c a l l y ,

I

k -

3=0 o < k < n.

Then, in f a c t ,

8111(1 a ls o a "bundle o f

Z

Gn-1 n_ 2' s

i s a bundle over

i s c a lle d the in c id e n c e correspon den ce, and

Z

pn * i t s e l f is

a f l a g m a n ifo ld . Now form the f i b r e p rod u ct

*

=

q

F x p Z: n

:

^ Gn,n-1

P F (i)

p

r

i s a f l a t morphism;

over

F

{1 ^ }

such th a t

P

-------------

n

in f a c t ,

in the sense th a t

F

p “ 1 (1 ^ ) s ^

*

i s a bundle o f

Gn_ 1 n_

adm its an open c o v e r in g x Gn-1 n_ 2 .

In p a r t ic u la r ,

dim * = dim F + dim Gn _ 1 n_ 2 = 2 + 2 (n -2 ) = 2 (n - 1) ■ dim Gn,n-1 and (ii)

*•

M oreover

q

i s a s u r je c t iv e morphism o f two n o n -s in g u la r

v a r i t i e s o f the same dim ension. an open subset dim ension (iii)

*

i s n o n -s in g u la r .

1

U C GR n-1 o v e r w hich

T h is im p lie s th a t th e re i s

c o n ta in in g a l l p o in ts o f co ­ q

i s f i n i t e and f l a t .

More g e n e r a lly , you can make any base e x te n s io n to o b ta in a s it u a t io n :

113

THE METHOD OP CHOW SCHEMES * x S v

P/ p x s

° n ,n - i

* s

•

One s t i l l has: p fla t q o f fin ite

T or-d im en sion

th e re e x i s t s open subset

U C Gn n-1 x S

c o n ta in in g a l l p o in ts o f d ep th 1, o v e r w hich T h e r e fo r e , i f

D C F x S

q

is

f i n i t e and f l a t .

i s a fa m ily o f cu rves o v e r

S, we can

form : $(D ) o

a c c o rd in g t o 1

= q *p *(D )

o

and 3 , L e c tu re 10.

The r e s t o f th e work c o n s is ts i n showing, as in L e c tu re 15, th a t $

is in je c t iv e ,

scheme

Y C P jj

i f and o n ly i f f o llo w s th a t

and th a t i f

D iv^ ^ h P^ ,

then th e re e x i s t s a c lo s e d sub­

such th a t an S -va lu ed p o in t o f

D iv^

i s i n th e image o f

the c o rre s p o n d in g p o in t o f P™- i s a p o in t o f Y . Then i t d Curvesg ^ hy. Even th e method i s s im ila r t o th a t o f L e c tu re

1 5 : one c o n s tru c ts an " in v e r s e " morphism: Y:

DiVg

A l l subschemesp

and then a p p lie s th e same c a t e g o r i c a l argument as in p a r t ( V I . ) , In 3ome sen se, th e d e e p e s t p a r t o f th e argument i s

L e c tu re 15.

th e same—the in v o k in g o f

th e e x is te n c e o f f l a t t e n i n g s t r a t i f i c a t i o n s t o v e r i f y th e h y p o th e s is in the c a t e g o r ic a l argument. An i n t e r e s t i n g c o r o l l a r y o f t h is approach i s the s tro n g e r f i n i t e ness theorem th a t i t

y ie ld s :

f i n i t e number o f elem en ts

f o r any g iv e n d eg ree

£ e Num (P )

d , th e re a re o n ly a

such th a t:

a)

d eg £ = d,

b)

£ i s r e p re s e n te d by a cu rve.

The e s s e n t i a l f a c t s behind t h is f i n i t e n e s s a re q u it e in t e r e s t i n g and u s e fu l. Whai: we want t o do i s

t o p ro ve c o m p le te ly a c l o s e l y r e l a t e d r e s u l t which

seems t o c o n ta in th e key p o in t , and w hich we w i l l use su b sequ en tly. THEOREM:

Let

Op(-D + d) P r o o f: Suppose

L C PR

is

D CF

be a cu rve o f d e g re e

spanned by i t s

We a re g iv e n an embedding

Then

F C Pn in d u c in g th e s h ea f

i s a lin e a r subspace o f dim ension

th e re i s a " p r o je c t io n " *:

d.

s e c t io n s .

( pn - L) -

P2

n -3 .

o ( 1) .

Then r e c a l l th a t

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE [ I n our

approach, we can d e fin e

Namely,

le t

h.

L =

*

H1 n H2 n H^,

c H°( Pn , 0 1 ) ) .

as a

F n L = 0, jt1:

a)

jr

is fin it e

T h e re fo re jt

then

F -

P2

Let

to a

morphism

fla t.

jrf i s a f f i n e because

i

denote th e in c lu s io n o f (i,

(i,

open s e ts and

p2" ^ i

jr“ 1( P2- j^ )

jt!

is

the r e s t r i c t i o n

jt 1)

F

in Pn »

i s an isom orphism o f

The f a c t th a t

F

P 2,* (£ p ) (where

co h eren t ( c f .

2 °, L e c tu re 7 ).

jt 1

Then

of

jtf

w ith a c lo s e d sub­

the d i r e c t image sh ea f

* i(O p )

is

i s i d e n t i f i e d w ith

the s tru c tu re s h ea f o f the image o f is

F

i n Pn x P2) .

T h e re fo re jrf

be a r e g u la r l o c a l r in g o f dim ension A-m odule.

n, and l e t

B

I f a l l lo c a liz a t io n s o f

w ith r e s p e c t t o maximal id e a ls a re Cohen-Macaulay r in g s o f dim ension B

be an B n,

i s a f r e e A-module. (C f. NAGATA, L o c a l R in g s , ( 2 5 . 1 6 ) , Now suppose th a t

such a morphism.

Then

D CF

Jt^.(D)

and EGA h ,

Computation o f deg S t a r t w ith a l i n e p o in ts o f the s e t

jt^(D) :

i s a curve o f d egree

xn )

=

d

and

Jt!

d

is 10.

to o .

iri(D ) :

ft C P2 w hich d o e s n 't c o n ta in any o f th e g e n e r ic ft n Supp jt^(D) i s 0-d im e n s io n a l, and

then

d eg jti(D ) (x ^ ...,

§ 15.M

i s d e fin e d by Norms, as in 2 °, L e c tu re

T h is i s a p la n e c u rve, and I cla im th a t i t s d eg ree i s

Let

T h is

is fin it e .

i s f l a t f o llo w s from the g e n e r a l r e s u l t :

A - a g le b r a , f i n i t e l y g e n e ra te d as then

=

Jt')

PR x P 2 ,

ju s t the same as

A

0 ( 1 ) have

of

.

the th re e fundam ental li n e s )

scheme o f

Let

h^

to a c lo s e d subscheme.

Since

Lemma:

P2.

jr.]

jtr e s t r i c t s

pg i s c o vered by a f f i n e

fa c t o r s :

d)

of

is a ffin e .

Pn ~

c)

and

is a ffin e :

(J ^ , j£2, * 3

b)

h 1, h2 and

Pn ~ L, and th ey d e fin e the p o in t

In p a r t ic u la r , i f

th a t it1

L )- v a lu e d p o in t

i s the h yperplane d e fin e d by

Then the th re e s e c tio n s

no common z e ro e s in

I c la im

p-

(

where

= (ft • jt i(D ))

ft n Supp jt^(D) .

.

A t each p o in t

x^,

le t

o^ = o ^

,

THE METHOD OP CHOW SCHEMES f\

e

H, R^ =

a l o c a l e q u a tio n o f

tio n o f

D

115 € Ri

in a n eigh borh ood o f the s e t

ge n e ra te d f r e e

-m odule, and

Hm(g^)

a l o c a l equa­

Then

is a fin it e ly

i s a l o c a l e q u a tio n o f

it^(D) .

M oreover, n

( i • *;(D)) = Yj djjnk - i / ( f i» 1611 S f

*

1=1 By an elem en ta ry r e s u l t on d e te rm in a n ts *,

dlmk

^

we g e t

8j_) = d3jnk Ri / ( f i» 8a.)

and, by d e f i n i t i o n : n I i= l

dimk

S i)

( « '* ( £ )

• D)

(o (1 ),

O p(D ))

=

d eg D d We now come t o the main p o in t : * !* ( * i ( D ) ) where

D’

= D + D*

i s e f f e c t i v e , b y statem en t ( * ) ,

sin ce the d i v i s o r c la s s o f (* i(D ))

i s a ls o

o (d ),

* * (D )

is

2 °, L e c tu re 101

o (d ),

And, in f a c t ,

th e d i v i s o r c la s s o f

hence th e d i v i s o r c la s s o f

D1

is

o^C-D + d) .

The theorem , t h e r e f o r e , w i l l be p roven i f we can show th e f o llo w in g (* )

/

F or a l l c lo s e d p o in ts

\

o f dim ension n-3 such th a t

j

th e d i v i s o r

v

through

D’ ,

x e P , th e re i s a lin e a r space L n F =

L

and such th a t

c o n s tru c te d as ab ove, does n o t pass

x.

In o th e r w ords, we r e q u ir e : * ! * ( * i ( D ) ) x = Dx F ir s t o f a l l ,

l e t 's

l o c a l r in g o f

P2

Let

g e R

m C R

* T:

Let

a n a ly ze what we need t o g e t t h is ou t: at

jt^ x ) ,

and l e t

be a l o c a l e q u a tio n o f

D

be th e maximal i d e a l such th a t

A

M -*■ M

.

R

a t a l l p o in ts Rm

be a 1 -d im e n s io n a l l o c a l r i n g ,

le t

be the s t a lk o f is

M

*1

o_ be the

* i(O p ) ( * 1( x ) ) ,

th e l o c a l r in g o f

F

a t 7r’ (x ) and l e t at

a f r e e A-module o f f i n i t e

an A - lin e a r i n j e c t i v e homomorphism.

Then:

le n g th (M /T(M )) = le n g th (A / (d e t T ) )

.

x.

ty p e ,

116

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

P a s s in g t o the co m p le tio n s , we fin d

R = R ® £ — The image o f Mn (g ) u n it .

in

( ^ 1)

each component

•

i s then the p rod u ct o f the Norms o f A

o .

to

® I

o th e r maximal id e a ls m1 C R

But we want

g

and

Mn ( g )

g

from

t o d i f f e r by a

T h e r e fo r e , f i r s t we need: a)

g

i s a u n it a t a l l o th e r l o c a l i z a t i o n s

t h is

of

R;

x' / x

such

ir 1 ( x 1) = ir 1 (x ) .

th a t

h o ld s , the image

T h e r e fo r e ,

Rm,

Supp (D) does n o t c o n ta in any p o in ts

i.e .,

If

o

= (V

Mn (g )

o

in

i s ju s t the Norm

from

(Rm)

o

to

secon d ly we can use /V

k)

£ —

i . e . , Rm

i s u n ra m ified o ver

o_,

th e map from the Z a r is k i ta n gen t space t o P2

Z a r is k i ta n gen t space t o If

t h is

h o ld s ,

Nm (g )

and g

at

Jt’ (x )

at

t o the

R^

What a re the c o rresp o n d in g g e o m e tric c o n d itio n s on a)

x

i s an isom orphism . /\ in Rm; t h e r e fo r e

d i f f e r o n ly by a u n it

th ey d i f f e r o n ly by a u n it in

or e q u iv a le n t ly F

L ?

C le a r ly

becomes: a ')

If and

L

is x,

n- 2

the lin e a r space o f dim ension

then

x

spanned by

i s the o n ly in t e r s e c t io n o f

L

L and

Supp (D ). On th e o th er

hand, lo o k a t the Z a r is k i tan gen t space

t h is c o n ta in s the ta n gen t space T£ gen t space

Tp

to

F, o f dim ension

induces an isomorphism o f T h e re fo re

T/T£

Pn

2.

at

x; ta n ­

M oreover, the f u l l p r o je c t io n P2

w ith the ta n gen t space to

The ta n gen t spaces t r a n s v e r s e ly a t

at

*

it(x ).

T£ and

Tp

to

L

and

F

in te r s e c t

x.

le t

M

be the 2 -d im en sio n a l lin e a r space through

space Tp

at

x.

The r e s t i s ea sy : w ith ta n gen t

F i r s t choose

r h (x ) = 0 \ h (y ) 7^ 0 , H

to

b) becomes: b’)

Let

T

L , o f dim ension n - 2 , and the

to

h € H °( Pn , 0 ( 1 ) ) such

x th a t

fo r y the g e n e r ic p o in t o f M or f o r y a g e n e r ic p o in t o f Supp ( D ) .

be th e co rresp o n d in g h y p erp la n e.

Second, choose

h T e H °( PR, c) ( 1 ) )

such th a t

r h '( x ) = 0 \ h '( y )

7^ 0 , f o r

y

th e g e n e r ic p o in t o f a g e n e r ic p o in t o f

M n H

or f o r

y

or f o r

y e (Supp (D ) n H) - { x } .

F n H

THE METHOD OF CHOW SCHEMES Let

H*

fiecL

be th e and

a 1'

space o f

L

of

c o rresp o n d in g h yp erp la n e. b 1)

and

L nF

Let

117

L =H n

i s 0 -d im e n s io n a l.

H1.Then

Let

L

s a tis -

L be a lin e a r sub-

dim ension n-3 n o t c o n ta in in g any o f the f i n i t e

set o f

p o in ts

L n F. QED The c o r o l l a r y o f th e theorem w hich can be used t o bound i n terms o f

deg (D)

is

x (O p (-D ))

t h is : If

D

i s a curve on

F,

then

(D • D) > -A • d eg (D )2 where

A = ( 0( 1 )

* 0( 1 )) - 2 .

We omit the p r o o f s in ce we have no o th e r a p p lic a t io n s f o r t h is f a c t .

LECTURE 17 GOOD CURVES In t h is le c t u r e , we want to g iv e a p a r t i a l answer to the t h ir d q u e s tio n posed in L e c tu re I :

What i s a good curve on our s u rfa c e

F ?

More

p r e c i s e l y , we d o n ’ t want to d is t in g u is h between l i n e a r l y e q u iv a le n t c u rves, so th e q u e s tio n becomes—what i s a

good d i v i s o r c la s s on

t h is :

G iven an a r b it r a r y i n v e r t i b l e

L (n )

should have e v e r y ’’good" p r o p e r ty one can ask f o r .

analogous q u e s tio n on a curve Then an i n v e r t i b l e 1

o

sh ea f

L

C on

sheaf

L e t ’ s be p r e c is e :

"g o o d ” i f i t s

fix

the induced i n v e r t i b l e

c la s s e s

P ic

has a f i x e d automorphism:

(I.) L

(II.) L

is

[hence

spanned by

c lo s e d p o in t

L *-► L ( 1) .

H ^ fL fn ))

= (o ) = (0 )

its

(IV .)

F C Pn

The f o llo w in g are

if

i

if

+ n = 0, i

+ n > 0, i > o ] .

s e c t io n s ; e q u iv a le n t ly , f o r e v e r y

x € F,

th e re i s a curve

D CF

such th a t

a L

1 x 4 Supp (D) L

la r g e enough.

s h e a f. Then the s e t o f d i v i s o r

ff^ L fn ))

, Op(D)

(IX I.)

d egree i s

L:

i s 0- r e g u la r : i > o,

the sh ea f

A ls o lo o k a t the

once and f o r a l l an embedding

£ ( 1) be (F )

The p o in t i s n

(C reduced and i r r e d u c ib le f o r exa m p le). C is

and l e t

v a r io u s good p r o p e r t ie s f o r

F ?

L, f o r v e r y la r g e

.

i s v e r y ample.

There i s a curve

D CF

w ith no m u ltip le components such

th a t Op(D) « L . What i s th a t i f a ll

the r e la t io n s h ip betw een th ese v a r io u s p r o p e r tie s ? L has any o f th ese p r o p e r t ie s ,

then

L (n )

Note f i r s t o f a l l ,

has th e same p r o p e r ty f o r

n > 0. P r o o f:

T h is i s

c le a r f o r

(I.) 119

and ( I I . ) .

F or ( I I I . )

we need:

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE LEMMA A: Assume

L

is

Let

L

and

spanned by i t s

M

be two i n v e r t i b l e

s e c tio n s and

M

sheaves on

i s v e r y ample.

F.

Then

L 0 M

i s v e r y ample. P r o o f o f Ternrna; a morphism

cp:

F -► P

S in ce

L

1 th e re i s a c lo s e d im m ersion

\|r:

th ese d e fin e a c lo s e d im m ersion

hand,

spanned by i t s

s e c t io n s , th e re i s

L s cp*(o(l)); sin c e M i s v e r y ample, ~ F -► P such th a t M s ^ * ( 0 ( 1 ) ) . T o g eth er m2

(cp, tyj) : On the o th e r

is

such th a t

Fx -+ P

x Pm2

one hasthe c a n o n ic a l

i

S egre immersion

ml m2+m1+m2

d

T h is i s d e fin e d by the req u irem en ts: 1

< 0 ( 1; j

= P ^ ( 0 ( 1; j

1

i*(Xj),

for

J

are the sections 0 < ^

(E x e r c is e : i

° (cp. t )

,

{

,

0 < j < m^g +

m] + m2 ,

P*(Xk) 0

0 < ^


for ^-n same order,.

check th a t t h is i s a c lo s e d im m ersion .)

I s a c lo s e d im m ersion o f [i

q* p 2 ( 0

o (q), t ) ] * ( 0 ( 1 ) )

F

in

Pw w w w ml m2+m1+m2

= (q>, t ) * ( p * ( o ( 1 ) )

T h e r e fo r e ,

and ® p*(o(1)))

= o

M

if

M.

i s v e r y ample.

la r g e enough n, say

(n , m) / (0 , 0 ). T h e r e fo r e ,

whenever

• M) i s p o s i t i v e f o r a l l v e r y ample sheaves

n > nQ,

M oreover,

L (n )

we saw in L e c tu re 17 th a t

w i l l be

v e r y ample, to o .

Then we

have a c o n t r a d ic t io n because (L n (- 1 )

• o(l))

= - (o(i)

• 0(1))

< 0

w h ile (L n (- 1 ) if

n

is

• L (n 0) )

= n (L • L)

- nQ( o ( l )

• o(1)) > 0

la r g e enough. QJ5D

G oing back t o the exam ples in L e c tu re F or

P 1 x P1, the p a ir in g on the 2-d im en sio n a l

13, we can check the r e s u l t . Num(F) 0 Q

i s g iv e n by the

m a trix

(i w ith one p o s i t i v e ,

I)

one n e g a tiv e e ig e n v a lu e .

p a ir in g on the 3 -d im en sion a l

Num(F) 0 Q

t(

_1 0 0 -1

V

00

For the second s u rfa c e , the

i s g iv e n by th e m a trix : 00 1

)'V

.

THE INDEX THEOREM One can p ic t u r e the s it u a t io n somewhat l i k e t h i s : Num(F) 0 R, and draw in the " l i g h t - c o n e ” (x

take th e r e a l v e c t o r space

• x)

=0.

Look a t the c lo s u re

o f the s e t o f p o s i t i v e r e a l lin e a r sums o f v e r y ample d i v i s o r c la s s e s :

I n term s o f t h is diagram , i t

i s u s e fu l to lo o k more c l o s e l y a t

the n u m erical c r i t e r i o n f o r v e r y ampleness i n L e c tu re 17: d e g (L ) > c 2

Let Let

X

x (L ) > “

€ Num(F) 0 Q

ftbe

be the image o f

the c a n o n ic a l i n v e r t i b l e

We use a d d it iv e n o t a t io n in Then u s in g

In f a c t , e,

L , and l e t s h ea f on

Num(F)

a)

d e g (x )

b)

d) + 2 X ( 0jj,) (X • X -
o 2 , > -

— 111— (h • h)

(x • h ) 2

.

I c la im t h a t , w it h a p o s s ib le m o d ific a t io n o f th e con sta n ts

b) i s im p lie d by the sim p ler c o n d itio n : b ')

(x

• x)

o (1) .

be i t s im age.

f o r p rod u cts o f i n v e r t i b l e

P r o p o s it io n 3, L e c tu re = (x

-------------2 (0 (1 ) • 0 ( 1 ) )

> — 1^2— " (h • h)

• (x

• h )2

.

c 2 and

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

130 P r o o f;

X

and suppose

In f a c t ,

le t

s’

be any p o s i t i v e number s m a lle r than

r

deg(x) > c

1

(x • x) >

L

Then I c la im th a t th e re i s a number

-e' — i— - — (x

f o llo w s im m ed ia tely

h)

•

A • (X

• h)

.

th a t b) h old s i f

A, use the f a c t th a t ( * )

curve f o r la r g e p o s i t i v e

.

A (In d ep en d en t o f X) such th a t

max { c 2, A ' s ( ^ ‘e , h) } To c o n s tru c t

2

(h • h)

|(X • a>) - 2X(Op) | < From t h is i t

e,

s a tis fie s

d e g (x )

is

la r g e r than

•

n\ i s re p re s e n te d by a

im p lie s

n (th e f i r s t P r o p o s it io n

o f t h is l e c t u r e ) ,

and

the f o llo w in g easy lemma: LEMMA: c^

G iven any i n v e r t i b l e

such th a t f o r a l l curves

|(Ojp(D)

P r o o f: ample;

then

Choose

(Op(D)

nQ

• M(nQ) )

the lemma f o llo w s i f

sh ea f

M

on

F, th e re i s a con stan t

DCF, • M) | < cM • deg D

such th a t and

(Qp(D)

M(nQ)

.

and

* M_1 (n Q) )

M_1 (n Q)

a re v e r y

a re p o s i t i v e , and

c^ = nQ. QED

COROLLARY:

There i s a p o s i t i v e e

such th a t i f

X e Num(F)

s a t­

is fie s a ")

d eg(X ) > 0 ,

b")

(x • X) >

— 1-15— =

then a l l i n v e r t i b l e

sheaves

(X • h )2 ,

(h • h) L

r e p r e s e n tin g

X a re ample.

Note th a t th ese c o n d itio n s sim ply d e fin e the p o s i t i v e nappe o f a cone in

Num(F) ® R.

On the o th e r hand, c o n d itio n s a) and b ’ ) d e fin e the

p ie c e o f t h is cone above a c e r t a in p la n e , i . e . ,

a tru n ca ted in v e r t e d cone.

Hence, the s e t o f v e r y ample sheaves in c lu d e s such a c o n e .* more r e s u lt which f i t s what i s sheaves?

in v e r y n i c e l y w ith t h is m odel.

the e x a c t shape o f the r e a l c lo s e d cone

There i s

one

The q u e s tio n a r is e s :

CQ spanned by v e r y ample

I t w i l l c e r t a i n l y alm ost alw ays be b ig g e r than the cone spanned by

the p o in ts s a t i s f y i n g our n u m erical c r i t e r i o n .

But a theorem o f Nakai and

M oisezon a s s e r t s :

* T h is , a t l e a s t , makes i t q u ite c le a r th a t i f L i s any i n v e r t i b l e then L (n ) s a t i s f i e s a) and b) f o r la r g e enough n.

s h e a f,

131

THE INDEX THEOREM If

L

i s an i n v e r t i b l e

s h e a f on

F , then

L

i s ample i f and

o n ly i f : a)

f o r a l l cu rves D C F ,

(c f.

(° p (D )

• L) > o ,

(L • L) > 0 ,

b)

K leim an, Am. J . M ath .,

cone spanned by

i 96 ^ ).

In our m odel, l e t

C

be the r e a l c lo s e d

th e i n v e r t i b l e sheaves Op(D)f o r e f f e c t i v e

P r o p o s it io n ,

t h is c o n ta in s the p o s i t i v e n u m erica l cone:

d e g (x ) > 0 .

Then N a ica i's theorem im p lie s th a t

cones w ith r e s p e c t t o th e i n t e r s e c t i o n p a ir in g ]

C

and

D.

By the

(x , x ) > 0 , CQ

a re Just d u al

LECTURE 19 THE PICARD SCHEME : OUTLINE Our n e x t o b je c t iv e i s

to p ro ve th a t the schemes

12 e x i s t .

Or, e q u iv a le n t ly ,

in v e r t ib le

sheaves o f n u m erica l typ e

P (t)

o f L e c tu re

t o p ro ve th a t th e re i s a u n iv e r s a l fa m ily o f In t h is l e c t u r e , we s h a ll make

some g e n e r a l remarks about the problem , and sk etch our method f o r s o lv in g i t . P r e c is e ly , p r e s e n ta b le . iso m o rp h ic: in v e r tib le

1 1 , |2

say

sheaves on

as f o llo w s : M

the problem i s

t o show th a t each fu n c to r

The f i r s t th in g to n o t ic e i s

g iv e n

M

P

on

th a t the fu n c to r s

a re two p o in ts in r e p r e s e n tin g

F x S

Num (P), and say and

|2 .

PicJ, P icJ ,

is r e ­ a re a l l

L] , L2

a re

D e fin e an isom orphism :

r e p r e s e n t in g an elem ent o f

PiC p . ( S ) , map

to M ® p * ( L 2 ® L " 1)

T h is r e p r e s e n ts an elem en t o f

PiC p (S )

.

and o b v io u s ly d e fin e s an isom or­

phism. The o n ly problem , t h e r e fo r e ,

1 = 0.

T h is fu n c to r w i l l be denoted

fu n c to r i s , P iC p (S )

( a f t e r G r o th e n d ie c k ).

in a n a tu r a l way, a group fu n c to r :

i.e .,

T h is

each o f the s e ts

i s a group and each map betw een then w hich i s p a r t o f th e fu n c to r ,

i s a homomorphism. te n s o r p ro d u ct. a scheme

i s t o r e p r e s e n t the fu n c to r f o r

PiCp

P (t )

Nam ely, m u lt ip ly two i n v e r t i b l e

sheaves on

F x S by

T h e r e fo r e , a c c o rd in g t o the g e n e r a l remarks in L e c tu re r e p r e s e n t in g

PiCp

i s a u to m a tic a lly a group scheme.

Th is

i s e s s e n t i a l l y G ro th e n d ie c k 's P ic a r d scheme.

[A c t u a lly , he tak es the d i s ­

j o i n t union o f the schemes r e p r e s e n t in g each

PicJ,,

P ic a r d scheme.

In the p re s e n t c o n te x t,

t h is i s a s i l l y

c o n s tr u c tio n :

and c a l l s t h is the

o v e r an a l g e b r a i c a l l y c lo s e d f i e l d ,

one sees the p o in t o n ly o v e r more com p lica ted

base schem es.] In f a c t ,

i t w i l l be more c o n ven ien t t o r e p r e s e n t

f i x e d , but v e r y ample

|.

Our method i s

the n u m erical c r i t e r i o n o f L e c tu re IT :

133

to choose one

|

PiCp

f o r one

which s a t i s f i e s

t h is gu aran tees th a t any

L

o f type

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

i

i s O -re g u la r and v e r y ample.

Then we s h a ll c o n s tru c t a s e c t io n

s

of

®: C u rves* —

£l c |

.

s If

d>

adm its ja s e c t io n

scheme

P (| )

P r o o f: s ° $

s,

then

P ic |,

Byh y p o th e s is

i s a morphism o f

$ o s i s the i d e n t i t y .

CurvesJ,

sub-

On th e o th e r

hand,

in t o i t s e l f which p r o je c t s th e whole fu n c ­

t o r onto a su bfu n ctor isom orph ic t o th e re e x is t s a p r o j e c t i v e s o $

i s re p re s e n te d by a_ c lo s e d

o f C (| ).

scheme

PicJ, .

C (|)

But we know from L e c tu re 15 th a t

r e p r e s e n tin g

Curvesj,.

T h e re fo re

i s in d ic e d by a morphism o f schemes: f:

D e fin e

P (| )

0( |) -

C( g )

.

as the f i b r e produ ct in th e diagram : P ( ! ) ------- ----- ►C (|)

| C (D — where

►c ( s )

X

C( e)

a is the diagonal morphism. Then

Hom(S, C( | ) ) p x f(P ) P (| ))

4—

o ,f)

in

Hom(S, P ( | ) )

such th a t

a

Hom(S, C (|)

(« )

x C (| ))

a re

t o the su bset o f

Curves|,(S)

hp^^

a re isom orp h ic.

P ic| ,

F in a lly ,

s in c e

c lo s e d im m ersion so

P (| )

a

i.e .,

the same.

i s isom orp h ic t o the subset o f and

a, P € ot x a and

i s isom orp h ic t o th e s e t o f p a ir s = (1 , f ) ( P ) ,

th e p o in ts T h is means

Hom(S, C (| ))

l e f t f i x e d by

th a t

l e f t f i x e d by

s ° ®.

Hom(S, f,

i.e .,

T h e r e fo r e , the fu n c to r s

i s a c lo s e d im m ersion, the- morphism

i s a c lo s e d subscheme o f

g

is a

C (| ). QFP

To c o n s tru c t sh eaf

L on

s, we must do the fo llo w in g :

F x S, o f typ e

f e c t i v e C a r t ie r d i v i s o r

|

a lo n g

D C F x S

M c P ic (S ). (a )

a L ® p*(M )

The c o n s tr u c tio n must have two p r o p e r t ie s :

i f we r e p la c e

L

by

we should g e t the same (b )

g iv e n an i n v e r t i b l e

c o n s tru c t a r e l a t i v e e f ­

such th a t

o ^ gC D ) f o r some

the f i b r e s ,

L ® PgCM’ )

f o r any

i t should commute w ith base e x te n s io n s

The keys t o our c o n s tr u c tio n a re the f o llo w in g then f o r any c lo s e d p o in t

x

€ F, l e t

Mf

e P ic (S ),

D,

i

:

sheaves: S -*■

F x

T -► S

.

g iv e n L

on

F x

Sbe the s e c t io n

S,

o

THE PICARD SCHEME: OUTLINE w hich maps

S

onto the c lo s e d subscheme

135

(x ) x S C F x S.

Then l e t :

M oreover, l e t g = p2 * (L ) ^f

.

Then th e re i s a c a n o n ic a l homomorphism

V fo r every over

x;

i.e .,

F x U,

a s e c tio n o f

hence a s e c t io n o f

Now r e c a l l th a t o f L e c tu re IT . we know th a t p a r t ic u la r ,

H1 (F , L ' )

an i n v e r t i b l e

= H2(F , L ‘ ) = ( 0 ) ,

the r e s t r i c t i o n o f g

g o v e r U C S g iv e s a s e c t io n o f * “1 i x (L ) o v e r U = i x (F x U ) .

L

g was assumed t o s a t i s f y the n u m erical c r i t e r i o n

T h e r e fo r e , i f

f o r e weknow th a t

g a lo n e .

« ^

L

sh ea f

t o any f i b r e

is lo c a lly fre e

Now suppose we choose any

L'

and th a t of

and th a t i t s r-1

on L*

F

is

o f typ e g,

i s v e r y ample.

p2

i s o f typ e

rank

In

g.T h ere­

r i s determ ined

by

x . , , . . . , x r _ 1 e F.

c lo s e d p o in ts

Then we have: ~ h =

^ > h : /t, i

r-i g ---- ► © i= i

hence

Ah:

A1*” 1 g —

AP " 1

©

i= i D u a liz in g ,

M

i

M i r-1 ® M i= l i

t h is g iv e s

^ r —1 (A h )*:---------M ^ ---- ► Hom(AP" 1 g, Og)

.

But

Hom(Ar_1 g, Og) s [i.e .,

AP g

g ® (Ap g ) ' 1

.

the can on ical p a ir in g o f ap_1 ( g ) and g in to the In v e r t ib le sheaf induces a homomorphism from g to Hom(a p~1 g, AP g ) , hence from

g ® AP g -1

to

Hom(a p~1 g , Og) .

I t i s c le a r th at th is i s an isomorphism].

P u ttin g a l l the in v e r t ib le sheaves togeth er

in cu rly brackets

t h is g iv e s a homomorphism: h' :

r-1 Og — g ® j ( A r g ) 1 ® [ L Mx. ] } 1=1

hence a g lo b a l s e c t io n :

a e r ( p x S, L ® p* | (A r g ) " 1 ® Suppose th a t Then

a = 0

a

\ L

does n ot v a n is h i d e n t i c a l l y on any o f the f i b r e s o f

d e fin e s a r e l a t i v e e f f e c t i v e C a r t ie r d i v i s o r

D C F x S

p2. such

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

[ r£

^ xS(D) = l ® p j | (Ar g r 1 ® w hich i s e x a c t ly what we want.

M oreover, i t

]

}

i s c le a r th a t a l l our step s

commute w ith base e x te n s io n , and th a t one winds up w ith the same D even * i f you r e p la c e L to s t a r t w ith by L ® p 2(M) . T h e re fo re our problem would be s o lv e d and

s

o

What does i t mean f o r p g 1( s )

?

a

would be c o n s tru c te d , p ro v id e d o n ly th a t

is h i d e n t i c a l l y on any o f the f i b r e s o f

Let Ls

does n ot van­

pg .

to v a n is h i d e n t i c a l l y on the

be the i n v e r t i b l e

sh ea f induced by L

fib r e

on t h is f i b r e , and

le t s (F )

L e c tu re 1 1 ) .

F, the p o in ts

p ” 1( s ) .

( r - 1 ) - t u p l e w hich works f o r e v e r y

i s n o t con ta in ed in any p ro p er T h e r e fo r e f o r alm ost a l l ( r - 1 ) cp(x1) , . . . ,

The d i f f i c u l t y , s.

cpC x.^)

w i l l be

how ever, i s t o fin d one

THE PICARD SCHEME: We w i l l n o t s o lv e t h is problem : such ( r - 1 ) - t u p l e e x i s t s . s t r u c t in g the s e c t io n on

F.

137

OUTLINE

in d eed , i t may w e l l be th a t no

In s te a d we s h a ll g e n e r a liz e our method o f con­ s.

N *r- 1

We s t a r t by ch oosin g a t o t a l o f

We group them in t o

N-1 s e ts o f

r

p o in t s , and one s e t o f

p o in ts r -1

p o in t s : { x 1 , 1 , x 1^2, . . . ,

x 1^r )

{ x 2 - , x 2 2, . . . ,

x

}

Grouping (r)

^XN -1 ,1 ' XN -1 , 2 '* * * ' x N - l,r ^ ( XN, 1 * XN, 2 9

> xN , r - l

For the l a s t r-1 p o in t s , make the same c o n s tr u c tio n as above, o b ta in in g : e H ° ( f x S, L ® p * { ( A r S ) ' 1 ® [ V

J j ) .

For each o f the o th e r s e ts o f p o in t s , how ever, we form h = ) h

:

g

-*•

1,1 hence

Ar & -► k . 9

T h is c o n tr a d ic tio n p ro v e s th e c la im . Now say b 1 i s d i s j o i n t from L. L e t b 1 = and. l e t H (i) be th e span o f a l l th e p o in ts Qqq, .,.. ... , eexxcceeppt t QQ^. ±. On On ththee ooththeer r hand, l e t q = dim L and choose q+1 p o in ts PQ, P j^, . . . , P^ from 911 w hich span L. L e t P* be any p o in t i n 911 o th e r th a n PQ, P ^^ .. .. . , o r P^. S ince th e Q 's a re in d e p e n d e n t n H (i) = Qf . i=0 i=o T h e re fo re , th e re i s an i , say i Q, such t h a t P*

bT =

I

4

H (iQ) . Now l e t

% +p *

i=o 1 ^3-0 3-^0 and l e t 9T* = 21’ » ! - P* + ^ . S in ce P* 4 H ((iQ) i Q) , 1=0

b* s t i l l c o n s is ts o f n+1

X

1k 3

INDEPENDENT 0-CYCLES ON A SURFACE in dependent p o in t s . S in ce

b-

But now

i s d i s j o i n t from

lin e a r space b ig g e r than

31* L,

L:

c o n ta in s

4

Q.

so

L.

dim L

PQ, P1, . . . ,

P^

and

T h e r e fo r e th ese p o in ts span a was n o t m axim al. QJED

COROLLARY: k (n + l)- 1 .

Let

31

be a s t r o n g ly s ta b le 0- c y c le o f d egree

Then f o r a l l c lo s e d p o in ts

th e re i s a d eco m p o sitio n ( 7 ) :

Q € Pn ,

k-1 a =

bi

1=1

where

b - j , . . . , b k-1

i s a c y c le o f Q

4

a re c y c le s o f

+ bk

n+1 in dependent p o in t s , and where

n in dependent p o in ts spanning a hyperplan e

b£

such th a t

H. P r o o f:

A p p ly the P r o p o s it io n to

31 + Q.

The r e la t i o n s h ip betw een th e two con cep ts o f s tro n g s t a b i l i t y i s

Let

F

be a n o n -s in g u la r p r o j e c t i v e s u r fa c e , l e t

be a g iv e n v e r y ample sh ea f on

X.-independent (w it h r e s p e c t t o spanned by i t s

x.-independence and

g iv e n b y:

P r o p o s it io n 3:

0( 1 ) F

H

F, and l e t

£ ( 1 )) .

Let

L

31

be a 0- c y c le on

be an i n v e r t i b l e

F,

sh ea f on

s e c tio n s and l e t

p X .(n + i)(d e g L ) then qj*( 31) i s a s t r o n g ly s ta b le o - c y c le on P . P r o o f:

If

H C Pn

i s a h yp erp la n e, then

i s a curve in the d i v i s o r c la s s o f [Number o f p o in ts i n

L.

cp*(H)

i s d e fin e d and

T h e r e fo r e :

cp*( 31 )

< [Number o f p o in ts i n

in

H]

31

in

Supp q>*(H) ]

< X, * [d e g q>*(H)} 2 = X. • (d e g L ) 2


x • (d e g | ) 2

suppose

|

We s h a ll, a t a l a t e r p o in t , put more c o n d itio n s on

"be th e c lo s e d im m ersion d e fin e d by

Then

f o r a l l c lo s e d p o in ts

x e F,

s ta b le b y P r o p o s it io n 3 o f the l a s t le c t u r e .

L

and i t s

dim C (g Q)

m • n

But

In s te a d , l e t

be one n u m erical typ e s a t i s f y i n g a l l th e

(* )

C

such

appears on one f i b r e .

w hich i s awkward.

in

D (g ) £ =

S.

Note th a t

Let

\

JKn y J le t

= H°(S, S ® K )

be th e co rresp o n d in g s e c t io n . I f th e s c a la r s s € S, then

£ a a

have th e p r o p e r ty th a t f o r a l l c lo s e d p o in ts

y 7

then the image o f And e v e r y

L

£ a y ay

( S ^ X ) K(s)

i s not zero ,

m eets a l l th e re q u ire m e n ts .

mutes w ith base e x te n s io n ,

Oy

a^

the image o f th e s e c t io n

£ a y ay

so i f is

L

is

F or the w hole c o n s tr u c tio n com_i th e sh e a f induced by £ on p 2 ( s ) ,

th e c o rresp o n d in g

occu rs o v e r some p o in t

have q u it e a b i t o f freedom :

s.

£ &y Qy

in

H °(F , L) 0 K.

On the o th e r hand, th e s e c tio n s

f o r e v e r y c lo s e d p o in t

s € S,

the images

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

148

0^

o f the

g e n e ra te th e v e c t o r space

ju s t a b o v e ).

(g 0 K ) 0 H ( s ) ,

(b y the C o r o lla r y

E v e ry th in g now fo llo w s from an ea sy lemma o f S e rre :

LEMMA ( S e r r e ) : a lo c a lly fr e e

Let

sh ea f o f rank

X r

be an (a lg e b r a ic ) on

X. L e t

scheme, and

V C H °(X , S )

be

le t

S be

a fin ite

d i­

m ensional v e c t o r space and assume: i)

r > dim X,

ii)

f o r a l l c lo s e d p o in ts

& > 0 H (x)

is

s e V

Then th e re i s an elem ent

x

e X,

the map from

V

to

s u r je c t iv e . whose image in e v e r y space 6

0 K (x ) i s

n o n -ze ro . P r o o f:

Let

N = dim V

C on stru ct a homomorphism

0 b y h (a 1, . . . , k e r n e l.

a^)

= Z a^e^.

Then ft

> Jt

H (x )

e 1, . . . ,

*

e^

be a b a s is o f

V.

T h is i s

o f any

0

Q)

s u r je c t iv e

i s l o c a l l y f r e e o f rank

the re s id u e f i e l d Tor

and l e t

h

by ( i i ) .

N -r : in f a c t ,

Let

ft

be i t s

te n s o r in g

w ith

x e X, we o b ta in :

( g , H ( x ) ) — ► ft H (x )

■■»

H (x )N— ■g ® K (x ) -

0

I (0) and

Tor ”

(ft, K ( x ) )

= (0 ).

Pass t o the d u a l e x a c t sequence: 0 -► Hom(S, ox ) — ►

Then

\

induces ( c f . EGA 2, (4 .1 ) P (x ):

Now

— — ►Hom(ft, o_x )

P [ Hcm(f t , o^.) ]

is

and ( 3 . 6 ) )

-► 0

a morphism:

P[H on(ft, ox ) ] — - P (o | ) l o c a l l y a p rod u ct o f

dim ension one le s s than the rank o f

.

= X

X x PN-1

.

w ith a p r o j e c t i v e

Hom(ft, £x ) .

space o f

T h e r e fo r e , by h y p o th esis

(i), dim P [Hom( ft, o^]

= dim X + N - r -

l < N - l

.

Look a t the com p osite: p2 o p (x ):

P[Hom(ft, ox ) ] -► PN-1

Because the dim ension o f th e domain i s s u r je c t iv e .

Let

le t

a^

is

a ^ ...,

a. e P^_-|

le s s than th a t o f

be a c lo s e d p o in t o u ts id e

be homogeneous c o o rd in a te s o f

the s o u g h t-fo r s e c t io n .

Suppose

Z aj_e j_

is

_a.

P ^ _ i>

it

i s n ot

Im (p 2 ° P ( x ) ) ,

Then I c la im th a t

and Z a^e

ze ro a t the c lo s e d p o in t

THE PICARD SCHEME: x e X.

Then ( a 1 J t. . , a N) i s in the s u b -v e c to r space

under the in c lu s io n t i o n a l on k ern el o f

\ .

T h e r e fo r e

Hom(ft, o-^) 0 H (x ),

a lg e b r a on b ra s :

CONCLUSION

Hom( ft, o-^) 0 K (x ) P

i.e .,

(a ^ ...,

a^)

ft 0 K (x )

H (x ) .

of

0 K (x ),

d e fin e s a l in e a r fu n c­

hence a homomorphism to

H9

P

from the symmetric

The maximal i d e a l

mx

and the

d e fin e a graded sh e a f o f id e a ls i n t h is graded sh ea f o f a l g e ­ a p o in t o f

lows im m ed ia tely th a t

P [Hom(ft, o ^ ) ] , p2 o ? ( \)

(c f.

L e c tu re 5, A p p e n d ix ).

maps t h is p o in t to

a,

It fo l­

w hich i s a c o n tr a ­

d ic t io n . QED

LECTURE 22 THE CHARACTERISTIC MAP OP A FAMILY OP CURVES We a re now re a d y t o a t t a c k th e e x is te n c e problem s r a is e d i n L e c tu re 2.

We s h a ll c o n s id e r f i r s t problem B.

t o d e fin e p r e c i s e l y th e " c h a r a c t e r i s t i c map" tu re 2:

t h is i s

p

A

and

B

The f i r s t ste p i s

in d ic a t e d ro u g h ly i n L e c ­

th e fundam ental lin e a r e s tim a te f o r f a m ilie s o f cu rves.

F i r s t some p r e lim in a r ie s : (A )

We w i l l need th e f o llo w in g e a sy c r i t e r i o n f o r r e g u l a r i t y :

P r o p o s it io n :

Let

£

be a n o e th e ria n l o c a l r in g , and

s u b fie ld isom orp h ic t o the r e s id u e f i e l d .

Then

o

k C £

a

i s r e g u la r i f and o n ly

if: fo r a l l f in it e

d im en sio n a l l o c a l k -a lg e b r a s

and s u r je c t iv e k-hamamorphisms

A,

AQ,

A -► AQ, the map

Homk ( o , A ) — Homk ( o , AQ) is

s u r je c tiv e .

P r o o f:

The c o n d itio n th a t

o

i s r e g u la r and th e c o n d itio n ( * )

a re b o th e q u iv a le n t t o the same c o n d itio n s on the co m p letio n T h e re fo re assume l o c a l r in g s ,

o

is

co m p lete, hence by s tru c tu re theorem on com plete

M oreover, we can assume th a t q.

i s r e g u la r

cp

fo rm a l power s e r ie s r in g s . o - h _ L i f t i t v ia

o_.

th e re i s a s u r je c t iv e homomorphism k[ [X 1, . . . ,

Then i f

_o_ o f

(* )

X j ]- ^

cpX.,, . . . ,

cpXQ

0

indu ces a b a s is o f

i s an isom orphism and one e a s i l y checks ( * ) C o n v e rs e ly ,

2 /m2 ^ _ k [ [ X l , . . . ,

s t a r t w ith th e homomorphism XjjU A

x,

,...,

t o homomorphisms:

* O

p m/m (m C o) .

\ — ► k i i x , , . . . , X J 3 A X , , . . . , x^)m m

1 51

Xn) 2

.

fo r

152

LECTURES

ON CURVES ON AN ALGEBRAIC SURFACE

Passing to the limit, one obtains a homomorphism:

.

o

But it is clear that \|r o q> is an automorphism of kttX^..., andsince cp is surjective, this implies that q> is an isomorphism, i.e., o_ is regular. QED (B) Suppose A is a finite dimensional local k-algebra. We will look quite frequently at the schemes F x Spec(A), so it seems worth­ while to put together at the outset the basic facts on their structure: i) As a topological space, F x Spec(A) is just F. only thing changed is the structure sheaf.

The

—FxSpec(A) is canonically isomorphic to o^ A. Namely, notice that the projections p1: F x Spec (A) -+ F, and p2: F x Spec(A) -► Spec(A) make o.FxSpec(A) into a sheaf of o^-algebras and a sheaf of A-algebras respec­ tively. Therefore, there is a canonical homomorphism: -F ®k A -*■—FxSpec(A)

*

But since, for affine open sets U C F, r(U, op ®k A)

= r(U, op) ®k A

and r(u> ^FxSpec(A)^ = r(U> ^

®k A »

(*) is an isomorphism of sheaves. iii)

Now let 1 = e^, e2,..., en be a basis of A where e2,..., en span the maximal ideal M. n —FxSpec(A) ~ -F +

over k, Then

Z ei ' -2f

1=2

and

n * * V ^FxSpec(A) = -F + L ei 1=2

11

= -F ‘ (1 + ^ 1=2

ei ' 2p)

Moreover, the truncated exponential sequence defines a homo­ morphism: n^ n ( 1 e± • oF)+ ► (1 + e. • Op

\

1=2

provided

\.

1=2

e*3 = o,

all

e e M,

p = char(k) .

THE CHARACTERISTIC MAP OP A FAMILY OP CURVES LEMMA:

The tru n ca ted e x p o n e n tia l i s alw ays an isom orphism .

P r o o f:

Use the tru n ca ted l o g t o g e t an in v e r s e .

We now come t o the main p o in t o f t h is le c t u r e : the f a m ilie s o f cu rves on by

I.

Not o n ly i s

I

P

over

a scheme o v e r

d e fin e s a c lo s e d im m ersion o f cu rves o v e r

I

Spec k [ e ] / e 2 . k,

it

Spec k [ e ] / s 2

but th e augm entation

k [e ] / e2

-► k

S p ec(k )

in t o

I.

I n t h is way, a fa m ily o f

d e fin e s e x a c t ly one o rd in a ry curve on

a v e c t o r p e r s o n ifie d :

t o in v e s t ig a t e

We den ote

P.

I

i t s e l f i s lik e

i s a s in g le p o in t w ith the s m a lle s t p o s s ib le

amount o f ” t a n g e n t ia l m a t e r ia l” s t ic k in g out in one d ir e c t io n . cu rves o v e r

I

i s b a s i c a l l y a curve on

F,

A fa m ily o f

p lu s an i n f i n i t e s i m a l deform a­

t io n o f t h is cu rve.

T h is i s an

F ix a curve

DCF.

D e fin itio n :

N^ =

in v e r t ib le

Q { £F (D ) } . —F

sh ea f on D,

shown to be the sh ea f o f germs

and i f

D

i s n o n -s in g u la r, i t

o f s e c tio n s o f the normal b u n dle.

can be Note the

e x a c t sequence: 0 -

P r o p o s it io n : f a m ilie s o f curves

P r o o f:

° p (D ) “* % “* 0

•

There i s a n a tu r a l isom orphism between the s e t o f

$ C F x I,

o f g lo b a l s e c tio n s o f

tc g iv e an

-

over

I,

w hich exten d

DCF,

and the s e t

N^.

To d e fin e a C a r t ie r d i v i s o r

open c o v e r in g

CU^} o f

F,

DC F x I

is

the same as

and l o c a l e q u a tio n s f o r

D.

In view

o f ( B ) , l o c a l eq u a tio n s a re o f the form : F.i

G± + e

• H.

where G± , H.

The induced curve on that this curve is

D.

F

£ r(U.,

op)

itself is defined by the first terms Recall that on

n Uj

Fj_ = (unit) • Pj

we must have: ,

or (Gi + eH^_) = ( &±3 + eb±j) • ( G. + e R . ) where a.. € r(U± n Uj, o£) L

e r(U. n Uj,

.

G^.

Assume

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE T h is g iv e s th e e q u a tio n s : Gi

= ai j

'

Hj_ =

+ ^

j Gj

hence % Hi 0 * “ G j = bi j But sin c e

G^

i s a l o c a l e q u a tio n f o r

and th ese e q u a tio n s say th a t T h is i s

D,

(H^/G^}

the s e c tio n co rresp o n d in g t o

' aj i

*

% /^i

J-3 a s e c tio n o f

o^,(D ),

p atch to g e th e r as s e c tio n s o f

N^ .

3).

Now suppose th a t w ith r e s p e c t to some open c o v e r in g s e ts o f l o c a l eq u a tio n s

F^,

gave the H

same s e c tio n s o f

s in c e

u n it

d^

Gj_

and

i n U^.

th a t e v e r y

D,^ / G .^ 1

is a

f o llo w s th a t

(G± + e ^ ) = (d ± + ec . hence the two d i v is o r s

*

a re b o th l o c a l eq u a tio n s f o r

Then i t

Then

H’

G7 " GT = c i 6 r (Ui ’ A ls o ,

two

N^.

3)

s e c tio n o f

and

3)'

• d± ) • (Gi l + e H ^ )

a re e q u a l.

d e fin e s a d i v i s o r

F in a lly , i t

3)

e x te n d in g

i s ea sy t o D

in t h is

check way. QED

COROIIARY 1 :G iven a fa m ily o f cu rves p o in t

s € S,

p: (where

Dg C F

3) C F

x S,

and a c lo s e d

th e re i s a c a n o n ic a l lin e a r homomorphism

i s the

r th e Z a r is k i tan gen t

-i

\ space

s }

Tg t o S a t

curve induced by

3 )).

E^F,

T h is i s the c h a r a c t e r is t ic map

o f the fa m ily . P r o o f:

g iv e n t e Tg ,

we have f :

w ith image

s

o b ta in a fa m ily

(c f.

I -+ S

L e c tu re kf A p p e n d ix ).

o f cu rves 3)f

C F x I

a c a n o n ic a l

Then, by base e x te n s io n

which exten d s

t io n ,

correspon ds to an elem ent

use the f u n c t o r ia l c h a r a c t e r iz a t io n o f th e sv e c t o r

on

Tg

€ H °(F ,

).

By

is

lin e a r ,

p (t)

Dg . To

f , we

the p r o p o s i­ show th a t

p

space s tru c tu re

(A pp en d ix, L e c tu re *0 , and check th a t t h is a g re e s w ith s tru c tu re we

have in tro d u c e d d i r e c t l y . COROLLARY 2: p

i s an

F or th e u n iv e r s a l fa m ily o f cu rves

isom orphism a t a l l c lo s e d p o in ts

s e C (| ).

3) C F x C (| ),

THE CHARACTERISTIC MAP OP A FAMILY OF CURVES P r o o f: of

t is

F o llo w in g the p r o o f o f the p re v io u s

alw ays isom orp h ic t o the s e t

of

f;

c o r o l l a r y , the s e t

and the

a e H °(F ,

set o f

Kp. ) i s isom orp h ic by the p r o p o s it io n to the s e t o f f a m ilie s 3)’ C F x I s e x te n d in g Dg . But by d e f i n i t i o n o f a u n iv e r s a l fa m ily , e v e r y 2)’ eq u a ls a

f o r a unique

f,

so th e s e t o f

3)’

and the s e t o f

f

a re isom or­

p h ic to o . QRD T h is would appear t o answer the fundam ental Problem B o f L ectu re 2.

But i n

fa c t, i t

does n o t.

We have o n ly g e n e r a liz e d the con cept o f a

fa m ily o f cu rves from the i n t u i t i v e v a r ie ty ,

one where the base i s a n o n -s in g u la r

to a "phony" one where the Z a r is k i ta n gen t space to th e base can

be huge, but the base can s t i l l be o n ly one p o in t I lem o f r e a l l y c o n s tr u c tin g f a m ilie s o f cu rves i s

The burden o f the p ro b ­

s h ift e d to the q u e s tio n o f

a s c e r t a in in g whether th e u n iv e r s a l base i s redu ced, or ( b e t t e r ) n o n -s in g u la r. Exam ple:

The f o llo w in g i s due t o S e v e r i and Zappa:

curve o v e r fit

k, and c o n s id e r v e c t o r bundles

&

le t

o f rank

C 2

be an e l l i p t i c

over

C

which

in t o e x a c t sequences: 0 -► o^ -► 6

By th e g e n e r a l th e o ry o f sh eaves,

—►Oq

—►0 .

such e x te n s io n s a re c l a s s i f i e d by elem ents

o f:

~ h 1 (c ' 2 c } But

H1(C , o^,)

i s a 1-d im e n s io n a l v e c t o r space; l e t

n o n -ze ro elem en t. s u r fa c e , i . e . ,

We take

F = P (g ) ,

le t

F

P, Q,

in t o a bundle o v e r

P

and

C

F

&

L e c tu re 5 ).

correspon d t o a T h is i s a r u le d

C

w ith f i b r e P 1.

be two d i s t i n c t p o in ts on

p ly in g by a s c a la r , at

(c f.

th e re i s a c a n o n ic a l p r o je c t io n it:

making

•

C.

th e re i s a unique fu n c tio n

Q, and no o th e r p o le s .

We can be v e r y

e x p lic it :

Up t o addin g a co n sta n t and m u lt i­ f

on

C

w ith sim ple p o le s

The c o v e r in g

C = (C - P)

U (C - Q)

- Up u UQ and

f e r(Up n uQ, Oq ) g iv e a 1-C zech c o - c y c le on

C

w hich r e p r e s e n ts th e g e n e ra to r o f

(up t o a s c a l a r ) . Then one can check th a t F = [P, x D p l

U [P ,

x UQ]

H1(C , o^,)

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE and th a t i f

tp

i s a c o o rd in a te on P1

second, then the

( t p ,

X )

x € Up n Uq , t p - t ^

=

P,

€

( t Q, x ) when

t^

X

Up

f. x Up

and

the f i r s t has l o c a l eq u a tio n

eq u a tio n

t ^ 1 , and

(# )

t p 1A

q1

=

1 - f

• t p 1,

(oo) x Uq

t p 1,

the

c o in c id e o ver

second has l o c a l

a u n it in a neighborhood o f (oo) x (Up n UQ)

C a ll t h is curve

E.

E i s a s e c tio n o f the morphism

an ir r e d u c ib le n o n -s in g u la r curve on °p (E )

*,

— t p * Qp

Ng = Og

in E

n ( p 1 x Up)

^ Og

in E

n ( p 1 x UQ)

.

and t h e r e fo r e i s

F , isom orph ic to

- tQ ' % T h e r e fo r e ,

one i n the

e P, x UQ

Now the curves g iv e n by ( » ) Up n Uq :i . e . ,

in the f i r s t p a tch ,

p a tc h in g i d e n t i f i e s the c lo s e d p o in ts

C.

in

P 1 x Up

in

P, x U Q

M oreover,

.

and the p a tc h in g

on the in t e r s e c t io n i s d e fin e d by the r e s t r i c t i o n to

t p V t Q 1.

t h is i s

By ( # ) ,

i , hence

Ngs Og

g l o b a l l y on

H °(F , Ng) s H °(E , Og) s k T h is

means th a t th e u n iv e r s a l fa m ily

c o n ta in in g

E

On the o th e r hand, i t .

i s ea sy to check th a t

e’

[ I t would f o llo w

i n the same component o f th a t the sh eaf

Og 0 O g (E ')

E

th a t o f

d , genus i e

cor­

as

a lo n e i s a com­ E'C

Fc o r r e ­

e , then

E D E’

was a d efo rm a tio n o f the sh ea f

E fl E 1,

and

E, Ng

of

Ngi but the

has a s e c tio n w hich

sin ce t h e ir E u ler c h a r a c t e r is t ic s are the same, t h is means

n E 1 = 0. ] E

O g(E r)

C g*1

would be a d efo rm a tio n , on

form er has a s e c tio n which va n ish es a t i s nowhere z e r o ;

e

For one can show th a t i f a second curve

sponded to a p o in t

th a t

o f cu rves o f d e g ree

E.

ponent o f Cg>1

O g (E ), hence

.

has a n o n - t r i v i a l Z a ri s k i-ta n g e n t space a t the p o in t

resp o n d in g to

= 0.

Cp’ 1

E of

E.T h e r e fo r e :

over

But a ls o th e d e g re e o f

C:

t h e r e fo r e

E’

over

C

must be

E 1 would a ls o be a s e c tio n o f

it

1

lik e

and would

have l o c a l e q u a tio n s : tp

Then

gp - g^ = f ,

= g p (x )

in

It-1 (U p ), gp e r (U p ,

t Q = 8 q (x )

in

« _1 (UQ) ,

and

f

g,-,

e r(U Q,

o j

.

i s a Czech co-boundary which i s a c o n t r a d ic t io n .

LECTURE 23 THE FUNDAMENTAL THEOREM V IA KODAIRA-SPENCER We a re now re a d y to p ro ve the theorem announced in L e c tu re 2, f o r which two a n a ly t ic p r o o fs were sketch ed. form o f t h is r e s u lt in the form D e fin itio n :

B

A cu rve

H1(O p (D ))

We w i l l p ro ve the s tr o n g e s t known

g iv e n a t th a t tim e.

D CF

-*■ H1(Np)

is

sem i- r e g u la r i f

i s th e zero-map .

THEOREM: ( S e v e ri-K o d a ira -S p e n c e r) . typ e

g.

Let

DQ

a)

c h a r(k )

b)

Dq

is

Let

A be a f i n i t e d im en sio n a l l o c a l k - a lg e b r a ,

A / i.

We must show

= 1,

i s n o n -s in g u la r a t

and l e t

open c o v e r in g

I

the c r i t e r i o n o f s e c tio n

th a t e v e r y curve D CF x Spec (A ) .

= i\ • A. of

fe r Uj

oF ®k A)

th a t th ese do n o t d e fin e a curve

a r b i t r a r i l y to

G1j €

h - Gio • pj

-

h^

eq u a l to

0.

of F^

D

A =

DQ a ls o dim I

in some a f f i n e

a r b i t r a r i l y to elem ents

. D

u n less

F^

andF j d i f ­ G. .

on lj

U. n 1

• pj n U j,

(o ^ (2) A ) * ) .

Then

1 • hij> h j e r ^ ± n V

and we must show th a t f o r a s u ita b le c h o ic e o f a l l th e

L e ctu re 2 2 .

exten ds

any ca se, th e re a re u n its

F± ■ Gu L ift

F^

To s t a r t w ith , l i f t

by au n it in U. n U .. But, i n —* -L J in (Op ® A) such th a t:

(A ),

an i d e a l and

C le a r ly we can a ls o assume th a t

F ix l o c a l e q u a tio n s F.

I C A

D C F x S pec(A ) which

F± e r ^ , The tr o u b le i s

If

s.

We s h a ll use

(U^}

be a curve o f

= 0,

C (g )

exten d s to a curve

DQ C F

5 € C (g ).

s e m i-r e g u la r ,

then

P r o o f:

Let

correspon d to the c lo s e d p o in t

F^

F i r s t n o te the i d e n t i t y :

1 57

and

G^j

we can make

1 58

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

n(hij + Gij ‘ hJk> = Pi ” GijFj + Gij(Fj" Gjk^k^ = Fi " GijGjkFk = 11 • hi k + (Gi k - Gli5G jk)F k

Let get:

and

F^.0^

denote the images of

h i ni,J + Gij Since Fk°^ gives:

=

is a local equation for hij T °) pi

(+ )

• h njk

L k w tb pr

p

G^j

h i ( nik + V DQ, and

^k

f 1 L

pi

and

Fk

in

.

o^,.

Then we

GijGJk \ . _(0) (i ) Fk f|0^ = Gp^

• Fj°h

this

G ljG jkG ik ’I

hence S h ±J J- i f > i s a 1-C zech c o - c y c le f o r the sh ea f £ is then

D

I -*all i,,

Np.

L e t t h is correspond t o

the o b s tr u c tio n to f in d in g

e x is t s .

In f a c t ,

DJ

# € H (N p ),

L e t 's check th a t i f

§ = 0,

suppose we make the changes: F{

=

F± + r)f± ,

Gij' =

Gij + 11

gij

Then one computes: n • V

-

F ,'

= Fi

Gi j ' - Gi j Fj

+ n • f ± - IfJ

■ ^ hij + fi S ince

g^

e q u a l to

i s an a r b it r a r y elem ent o f 0

fo r a l l

i,

j

hi j by a s u ita b le c h o ice o f

+ q

- fJGij " W

r(U .

G ij - i P j g i j

•

n U j, O p),

{f^ }.

- f j Gi j ) e ( f S0 ) )

we can make

h^ .1

.

But t h is means th a t

h.

or th a t

•

i f we can make

.

- r fo ) Fi

e x is t s i f

•V

=

Z &) “ Fi

f . Fj

f , (mod ^ f }

- $ i s a Czech co-boundary in the sh ea f £ = o. Now by h yp o th e s is b ) , the homomorphism H1(Np) - 1 *

H2(Op)

coming from the e x a c t sequence 0 -► Op

Op(D) -*■ Np -*■ 0

Np.

» T h is p ro ves th a t

D

THE FUNDAMENTAL THEOREM V IA KODAIRA-SPENCER is in je c t iv e . th e s e c tio n s O p (D ), i t

T h e r e fo r e i t V

Fi 0) o f

d (§ ) = 0 .

s u f f i c e s t o p rove th a t Op(D)

lift

f o llo w s from form u la ( t )

But s in c e

the c o -c h a in r e p r e s e n tin g

th a t

d( £ )

§

in t o

i s re p re s e n te d by the Czech

2 c o - c y c le : 1 - Gl j

• G 1 k • GZk

"ijk But

i s an o b s tr u c tio n t o l i f t i n g

(Op ® A )*

t o a c o - c y c le i n

may choose

}

( o^ ® A ) * :

such th a t

0^

the 1- c o - c y c le i n fo r i f

it

= Gi k ,

(G-^j}

can be l i f t e d ,

i.e .,

ffi j k

= °*

in then we E v e ry th in g

fo llo w s now from : LEMMA:

(Op ® A )* -► (Op ® A )* — 1

s p lits .

P r o o f:

One m e re ly uses the e x p o n e n tia l, as the c h a r a c t e r is t ic

0:

is

(Op ® A)

------------------ ► (Op ® A)

a

a

(1 + Op ® M) -------- -

(1 + Op ® M)

2,|exp

ai exp

(Op ® M) +

(O p ® M ) + ------------ ►

Now s in c e

M -► M

s p l i t s as a s u r je c t io n o f v e c t o r sp aces,

s p l i t s as a s u r je c t io n o f sheaves o f a b e lia n grou ps.

Op < S> M -► Op M

T h is p ro v e s the lemma. QED

COROIIARY: Then

5

is

Let

D C P

s a t i s f y the h ypotheses o f the

co n ta in ed i n o n ly one component dim Z P r o o f:

=

Z

of

theorem .

C( |) and

dim H °(P , Np) .

S ince th e l o c a l r i n g

o^

of

C (|)

at 8

is

r e g u la r

dim Z = dim o^ = dim T& = dim H °(P , Np) by C o r o lla r y 2 o f L e c tu re 2 2 . T o 'p r o p e r ly understand t h is theorem , i t req u irem en t o f s e m i- r e g u la r it y i s v e r y w eak. by the q u it e p a t h o lo g ic a l cu rve

E

should be added th a t the

O f co u rse, i t must be v i o l a t e d

i n th e example o f L e c tu re 2 2 ;

but a 1 -

r e g u la r curve i s a s e m i-r e g u la r , and we know th a t f o r e v e r y i n v e r t i b l e L

on

P , th e re i s an

mQ

a re 1 - r e g u la r i f

sh eaf

such th a t a l l cu rves w ith g lo b a l eq u a tio n s in m > mQ.

L o o k in g back a t the exam ples o f L e c ­

tu re 1 , i t w i l l be seen th a t a l l the cu rves n o t d e s c r ib e d as superabundant a re 1 - r e g u la r , hence s e m i- r e g u la r . where

P

M o reover, lo o k a t th e analogous case

i s r e p la c e d by a n o n -s in g u la r curve

7

and

C (| )

i s re p la c e d by

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE C (d )—the u n iv e r s a l fa m ily o f 0 - c y c le s on 7 o f d eg ree

D C 7

is

s e m i-re g u la r sin c e

H1 (N p)

In f a c t , as i s w ell-k n ow n ,

C (d )

d.

Then e v e r y 0 - c y c le

has 0 -d im en sio n a l su pport, hence

Np

is

= (0 )

ju s t the

.

d - t h symmetric power o f

7

which i s n o n -s in g u la r . On the o th e r hand, the requ irem en t o f c h a r a c t e r is t ic c e n t r a l.

0

i s q u ite

For the l a s t fo u r le c t u r e s we s h a ll t r y to g e t c lo s e r to the h e a rt

o f the theorem so as t o b r in g out s e v e r a l ways in which the c h a r a c t e r is t ic r e s t r i c t i o n can be " e x p la in e d ." what the p r o o f r e a l l y does i s

To see what should come n e x t, n o te th a t

to reduce the l i f t i n g problem f o r

problem f o r i t s

a s s o c ia te d i n v e r t i b l e

c o - c y c le

Then why n o t e lim in a te

j .

p rove the theorem in form sheaves?

sh ea f D

2 px spe c ( A ) ^

D

to the

d e fin e d by the

e n t i r e l y from the problem , and

A o f L e c tu re 2 - e n t i r e l y in terms o f i n v e r t i b l e

2k

LECTURE

THE STRUCTURE OP 0 1°

In t h is le c t u r e we want to put to g e th e r our whole s e t-u p :

in L e c tu re 15, we co n s tru c te d the schemes L e c tu re 2 1 } we co n s tru c te d the schemes

C (£ )

P (£ )

p a r a m e tr iz in g c u rves;

p a r a m e tr iz in g i n v e r t i b l e

in sheaves.

The morphism o f fu n c to r s D h- o(D ) in du ces a fundam ental morphism o f schemes ®:

C (£) -

P (£ )

In L e c tu re 13 we d e s c r ib e d the f i b r e fu n c to r s rep i’ esen ted

Curves^

and

COROLLARY: if

the

sh ea f

L

P ic ^

$

a re p r o j e c t i v e

o ' 1 (X ) s P [H ° (L ) ] ®

C (£ )

spaces.

In f a c t ,

.

can be d e s c r ib e d somewhat s i m i l a r l y ( c f .

d ie c k * s Bourbaki t a l k , expos§ 2 3 2 , p . fo r d iffe r e n t

now th a t wehave

\ € P ( £ ), then c a n o n ic a lly :

correspon ds t o

The g lo b a l s tru c tu re o f

L in S ys^ :

we g e t the C o r o lla r y :

The f i b r e s o f

on F

.

£*s, th e schemes

1 1 ).

P (£ )

a re a l l

o v e r them are v e r y d iffe r e n t > - fo r

d e g (£ )

and i t s

< 0, th ey a re empty;

e x p l i c i t d e s c r ip t io n

T h e r e fo r e , we o n ly g iv e the r e s u l t Let

(* )

U C P (£ )

For exam ple, i f

F

D CF

correspon ds t o th e p o in t P ( £)s a t i s f i e s

(* ) (in

is a

r e q u ir e s some te c h ­ of3

,

i n a s p e c ia l ca s e .

be g iv e n such th a t: x e U,

if

1^ i s

the i n v e r t i b l e

c o rresp o n d in g t o

x,

then

H1(F , L^)

f o r a l l c lo s e d p o in ts sh ea f on

fo r

£ ,0

For some

n i c a l con cep ts coming out in the fu r t h e r developm ent o f the th e o ry L e c tu re 7.

th a t,

iso m o rp h ic, whereas the schemes

d e g (£ ) ” *■ + 00, th ey in c r e a s e i n d e f i n i t e l y in dim ension. f a i r l y c o m p lica ted f i b e r i n g ,

G rothen-

The i n t e r e s t i n g th in g i s

i s a curve f o r w hich

5 e C (£ ),

H1 (F , £ p (D ))

then some neighborhood

= (0 ).

= (0 ), U

v i r t u e o f the r e s u lt s o f 3 °, L e c tu re 7 ).

1 61

of

and i f $ (5 ) €

D

162

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE P r o p o s it io n :

There i s a l o c a l l y f r e e

sh ea f

&

on

U,

and a

commutative diagram : ® '1(U ) st P( S )

C (| ) 3

P (l) 3 U

F x U.

P r o o f:

Let

L

A b b re v ia te

p2 :

F x U -*■ U

be the u n iv e r s a l fa m ily o f i n v e r t i b l e

i s a l o c a l l y f r e e sh ea f on

U,

sheaves on

to

p.

A c c o rd in g t o L e c tu re 7, 3 , P * (L )

and i f

g:

T

U

i s any morphism and i f

one

makes the base e x te n s io n : F x T •

F x U

g then

q * (h ( L ) )

p#( L ) ,

s g (p * (L )).

Now l e t

U 6

be th e d u al l o c a l l y f r e e sh e a f to

i.e ., 6 = HQTCo^ ( P * ( L ) , 0^)

We s h a ll now p ro ve th a t

4- 1 (U)

and

p( S )

a re iscm orp h ic o v e r

same method used in L e c tu re 13 t o show th a t je c tiv e

space:

L in Sys^

U

by the

i s re p re s e n te d by p r o ­

we s h a ll g iv e an isomorphism betw een t h e i r fu n c to r s o f p o in t s .

More p r e c i s e l y , g iv e n a T -v a lu e d p o in t

g:

T -► U

of

U,

n a tu r a l isomorphism between th e s e t o f T -v a lu e d p o in ts o f and the s e t o f T -va lu ed p o in ts o f w i l l be f u n c t o r i a l i n

P ( &) o v e r

g.

we s h a ll g iv e a $” 1(U)

over

g

S ince t h is isomorphism

g , the theorem w i l l be p roven .

But p roceed in s e v e r ­

a l s ta g e s :

{

s e t o f T -va lu ed p o in ts

)

of

j

(U) o v e r

g

( s e t o f f a m ilie s o f cu rves ^

)

( M T h is fo llo w s by the f u n c t o r i a l d e f i n i t i o n o f

(ii)

)

on

T,

and o f

C (|)

D C F x T

T , and s e c tio n s o f

on

M

opxT(D)

s

sheaves

on

h * (L ) ® q*(M )

F x (t)

over

T.

h * (L ) ® q*(M )

T h is f o llo w s because

D

is

ju s t a r e l a t i v e C a r t ie r d i v i s o r o v e r T * * h (L ) 0 q (M) ;

g lo b a l e q u a tio n i s a s e c tio n o f a sh ea f o f the form a r b it r a r y C a r t ie r d i v i s o r on

F x T

whose and an

i s a r e l a t i v e C a r t ie r d i v i s o r i f i t s

g lo b a l eq u a tio n i s a n o n -ze ro d i v i s o r in each f i b r e o v e r n o n -ze ro th e r e .

M

in d u c in g n o n -ze ro s e c tio n s in each I fib r e

T,

.

$.

set o f in v e r t ib le

such t h a t , f o r

sh ea f

Ojjyp(D) a h * (L ) ® q*(M )

s e t o f f a m ilie s o f cu rves some i n v e r t i b l e s h e a f

D C F x T

such t h a t, f o r some i n v e r t i b l e

T, i . e . ,

if it

is

THE STRUCTURE OP

a o f h * (L ) ® q*(M )

But a s e c t io n as a s e c t io n

t

over

4

T

163 over

F x T

i s th e same th in g

of

q*(h*(L) ®q*(M)) s q#h*(L) ® M s g * (p * (L )) ® M .

a

M o reover, the c o n d itio n th a t fib r e

over

T

is

should induce n o n -ze ro s e c tio n s on each

th e same as th e c o n d itio n th a t

should have a n o n -ze ro

t

image in { g * [ P * ( L) ] ® M) 0 f o r a l l c lo s e d p o in ts

t € T.

th in g as a homomorphism

K (t )

But a s e c t io n

Ham (g * (p * L ), lip

o j

g [p * L ] M

i s th e same

——-*■ M

i.e .,

g iv e n a homomorphism from

g (p * L )

® M,

one g e ts a s e c t io n o f

t h is i s an

t

of

t

h:

M:

i s e q u iv a le n t to th e c o n d itio n th a t Ham0

to h

£T h.

and a s e c tio n o f

g [p * L ]

M o reover, the c o n d itio n on

be s u r j e c t i v e .

F in a lly ,

sin c e

o ^ ) = g*[Ham0^ (p * L , o^j) ]

(g * (p * L ),

« g*6 we g e t : set o f in v e r tib le

sheaves

M

on

(iii)

fib r e

F x {t)

over

( s e t o f i n v e r t i b l e sheaves

T ,^

and s e c tio n s o f h ~ (L ) q*(M ) in \ n o n -ze ro s e c tio n s i n each

(

on

) M

T, and s u rje r je cclt io n s

«:

g * (S ) - M

.

T.

But by the Appendix t o L e c tu re 5, t h is l a t t e r s e t i s isom orp h ic t o the s e t o f T -v a lu e d p o in ts o f

P( & )

lift in g

th e g iv e n T -v a lu e d p o in t

g

of

U.

T h is g iv e s the s o u g h t-fo r isom orphism .

2 i.e .,

N ext we want to d e s c r ib e the i n f i n i t e s i m a l s tru c tu re o f P ( | ) ,

i t s I- v a lu e d p o in t s ,

s i c a l l y on we c a lle d

F.

ju s t as we have d e s c rib e d th ose o f

We may as w e l l lo o k a t th e case

P (t )

b e fo r e .

P (t )

ous in th e f o llo w in g sen se: i s an automorphism

T

of

if P (t )

% = 0:

i s a group scheme, and co n seq u en tly homogene­ x, y

a re two c lo s e d p o in ts o f

such th a t

T (x )

= y.

m e t id a t e ly im p lie s th a t a l l t o p o l o g i c a l components o f th a t th e y a re a l l isom orp h ic t o each o th e r ; p on en ts; and th a t

P ( T) r e ^

L e c tu re 11,

scheme i t s e l f ,

(V ). ]

In f a c t ,

u s in g Remark (V )

P (t ) ,

T h is , i n i t s e l f , P (t )

th e re im -

a re i r r e d u c ib le ;

th a t th e y have no embedded com-

i s n o n - s in g u la r .

the f a c t th a t th e re i s an open dense su bset l a r —c f .

C (|) —i n t r i n ­

t h is i s th e scheme

[The l a s t by hom ogeneity and U C P ( T)p ed

P ( T) r e a

o f L e c tu re 11.

w hich i s n o n -sin gu ­

i s e a s i l y checked t o be a group A ls o the component o f

P ( T) r e a

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE c o n ta in in g the i d e n t i t y v a r ie ty o f

e

i s a group scheme:

Because

P (t )

the c l a s s i c a l P ic a rd

i s a commutative group scheme, f o r any

th e re i s even a c a n o n ic a l automorphism la r ,

t h is i s

F.

T

such th a t

T (x )

= y.

x, y In p a r t ic u ­

th ese automorphisms g iv e c a n o n ic a l isomorphisms o f the Z a r is k i tan gen t

spaces a t a l l the c lo s e d p o in ts o f

P (t )

w ith each o th e r .

T h e r e fo r e , we

may l i m i t o u rs e lv e s t o c o n s id e r in g the I- v a lu e d p o in ts o f l y i n g k -va lu ed p o in t i s

the i d e n t i t y

0.

P (t )

whose under­

Use the tru n ca ted e x p o n e n tia l s e ­

quence :_________________________________________ _______ 0

where

a (f)

= 1 + e • f

a ls o a sh ea f o f

£p—

(c f.

o£_-r ------►o£

► 0

—F x l

L e c tu re 22,

(B )).

T h is s p l i t s

o ^ -a lg e b ra s v i a the p r o je c t io n

p 1:

s in c e

F x I -► F.

Opyj i s

T h is g iv e s

the diagram o f groups: -

H1(o p ) P ic (F ) l\\

Group o f I- v a lu e d p ts . a t

0 e P(t) .

r Group o f I- v a lu e d 1

I" Group o f k -va lu ed

L p ts .

L p ts .

of

II P( I )

J

I n o th er w ords, the Z a r is k i ta n gen t space 1

i s c a n o n ic a lly isom orph ic t o

H (F , o^,) .

a l l y an isom orphism o f v e c t o r spaces.

Tn

of

] -

H P (0

a t th e i d e n t i t y

One must check th a t t h is i s a c tu ­

T h is i s

l e f t t o the r e a d e r :

it

can

be done v i a the methods o f the Appendix t o L e c tu re 4.

3 $ (5 ) .

Now suppose th a t

The morphism

(# )

$

r Z a r i s k i ta n g e n t"] 0

space to f i b r e at 6

5

i s a c lo s e d p o in t o f

C (| )*

Let

^ =

induces an e x a c t sequence o f v e c t o r spaces: r Z a r i s k i ta n g e n t"] -► _

space t o _at

C (|)

5

,

*

_

We want to in t e r p r e t t h is whole sequence i n t r i n s i c a l l y on

r Z a r is k i ►

_ at F.

tan gen t

space t o

P (| )

X .

But lo o k a t

the e x a c t sequence o f sheaves: °F where

D CF

is

Op(D)

the curve co rresp o n d in g to

quence o f v e c t o r spaces: ( o,-,(D))

%

0

T h is d e fin e s the e x a c t se-

THE STRUCTURE OF n irfN p j)

a)

T z a r is k i ta n gen t

165

$

~| by L e c tu re 22

= [_space to C (| ) a t s j

H1 ( a-,) -

b)

T Z a r is k l tan gen t I space t o P (| )

and

by the automorphism

*1

0 by 2 ;

a t XJ T of

U

P (| ) ta k in g

0

to

X

I i.e .,

T

t r a n s la t io n by

and

d i n ('#) and (# ) ’ a re

under th ese i d e n t i f i c a t i o n s o f th e v e c t o r spaces. Check o f c o m p a t i b i l i t y :

G^

X.

The homomorphisms 0*

P r o p o s it io n : th e same

is

in a f f i n e

open s e ts

{ U ^ }.

Let

D be

d e fin e d

Any s e c t io n o f

by l o c a l eq u a tio n s

i s d e fin e d by d a ta :

Oi, -Hj_ e r(U±, 0^)

H ^ , where

- Hj /Gj and the c o rresp o n d in g curve

2)

in F.

Then

th e i n v e r t i b l e sh e a f

e r(U± n nj(

F x I

is

= G± + eH i

®( 3) ) =

^ ,)

,

g iv e n by l o c a l e q u a tio n s : .

$ ) is

d e fin e d

b y th e 1 Czech c o ­

c y c le

on

F x I.

S in ce

T h is i s

(G^G-j1}

computed out a s: = (Gi

+ eHf

- ■ [

i s a 1 - c o - c y c le d e f in in g

* (Gj 1)

' *

O p(D ),

the I- v a lu e d p o in t term .

• (1 - sHj • G j1) • i.e .,

{a . .} back to the o r i g i n i n II P ( l ) 1J . 5 T h is g iv e s the 1- c o - c y c le

X,

one t r a n s la t e s

by d i v i d i n g by t h is

[ ' • • ( i - y ] w hich i s

the image under th e tru n c a te d e x p o n e n tia l o f th e 1 - c o - c y c le *

in

o . Then P the o th e r hand, of

{ t .

.}

is

- r 5 k - 5 n ' G_. G. '

1 ,

J______________________ _

H ( 0- ) co rre s p o n d in g t o ®( D ) . On -t* i s c e r t a i n l y th e coboundary o f the s e c t io n {H^/G^} the p o in t o f

Np. OLD

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

166

The f i n a l i d e n t i f i c a t i o n i s v iz .

th a t i f

H °(F , L)

L

i s an i n v e r t i b l e

correspon ds t o the cu rve

l e f t to the re a d e r t o c a r r y ou t:

sh ea f on D,

F , and i f

the s e c tio n

hence t o the c lo s e d p o in t

s e 8

in

the lin e a r system P = P [H ° (L )] of

L,

then the Z a r is k i ta n gen t space t o

p

p h ic t o : H °(P , L )/ k • s

at

8

is

c a n o n ic a lly iso m o r­

LECTURE 2 5 THE FUNDAMENTAL THEOREM VIA GROTHENDIECK-CARTIER Suppose

D C F

is a curve of type

t, that

D

5 € C(|), that Ojj,(D) corresponds to X € P(|), and that H 1 (F, L) = (0), then the following are equivalent:

neighborhood the form

i)

p(|)

is non-singular at

x,

ii)

C(|)

is non-singular at

5,

iii) iv)

C(|) P( 5 )

is reduced at is reduced at

x U.

P( t )

8, x .

This implies that i) and ii) are equivalent, and that iii) Naturally, i) implies iv) .

is isomorphic to

hence

If

Proof: By the results of 1 ° of the last lecture, there is a U of X e P(|) such that the subset ~1(U) of C(|) is of

and iv) are equivalent. P(|>

corresponds to L = oF (D) .

P( t ), and

But conversely, since

is a group scheme, if

P( t )

P(|)

and

is reduced, then they are both non-singular (2°, Lecture 2 k ) . In characteristic 0, these conditions always occur because of: THEOREM 1

over

k. If

Proof: at

e.

(Cartier) : Let

char(k) = 0, Let

v

then G

G

be a (algebraic) group scheme

is non-singular.

be the completion of the local ring

Og

of

G

Multiplication is a morphism G x G

such that

ii(e x e) = e: *

11 :

v

-+

therefore

— ii

G defines a homomorphism

[completion of

A

o_ _] s u 0 -exe k v

where 0 is the completed tensor product [i.e., use the fact that Oexe is the localization of 0 with respect to the maximal ideal (c^ 0 m@ + me ® ^e) ^

since

v

is a group law, the restriction of

n

to either

G x (e) C G x G or (e) x G is just the identity from

G

to

G.

C G x G Algebraically, this means that if you 167

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE map

d

fie ld u.

v

0^.

v

onto

by mapping e i t h e r o f the two f a c t o r s onto i t s re s id u e

k, and compose t h is w ith

fi*, the r e s u l t i s

*

/ X-

fi(a ) must go t o

0

if

the i d e n t i t y from

a € m, the maximal i d e a l o f

T h is means th a t i f

v

to

u, then

1 0 a - a 0 1

v 0^. v

e i t h e r fa c t o r o f

i s mapped onto i t s re s id u e f i e l d ,

i.e ., (a )

| i ( a ) e 1 0 a

+ a 0 1 + m 0 m . k

We now p ro v e : (* )

f o r a l l lin e a r fu n c tio n a ls

v

D:

v

a n n ih ila t in g

P roof o f (* ) : th a t

F = 0

on

k

Extend

and on

Then

D

is

D

F: u -*■ k

by r e q u ir in g

be the co m p o sitio n : ►v 0 k = k

a n n ih ila t e s

k.

u

M oreover, by the e x p r e s ­

a € m, D (a )

= 1 0 F[ 1 0 a + a 0 1 + (R) ] , = F (a )

But

D

f.

t o a l i n e a r map

Let

c l e a r l y lin e a r and

if

f

m/m2 -*■ k, th e re i s a d e r iv a t io n

and in d u cin g

u — -— ► i) 0 v k

D:

s io n ( a ) ,

2

m .

f :

k

(1 ® F) (R) € m,

+ (1 ® F) (R ) hence

R € m®

m

.

D indu ces

f as

a map from m/m2

to

v/m = k.

I t rem ains t o check: D (a • b) if

a , b e m.

= a • Db + b • Da

But ju s t compute: fi* (a

• b)

= n * (a )

• fi* (b )

= (1 a + a ® 1+ R) = (a 0 1)

• (1®b + b®1

• |i*(b) + (b 0 1)

+[ l ® a b - a b ® l = (a 0 1) /s R, S e m 0 m,

where

D (a • b )

and

• n * (b )

^ p T e u 0 i + u 0 m

= ( i ® P ) [ (a ® 1) = a • [(1

+ R * I ® b

+ (b 0 1) .

+ S)

• n * (a ) • n * (a )

+S*

l ® a

+ R * S

+T

T h e r e fo r e ,

• n*b + (b ® 1)

• u*a + T]

® F)|i*b] + b • [(1 ® F )n * a ]

= a • Db + b * Da To com plete the Let of

f 1, . . . ,

v.

fn

W r it in g

p r o o f o f the theorem , l e t

X1, . . . , Xn

be a b a s is

of

m/m2 .

be a d u al b a s is , and exten d th ese to d e r iv a t io n s D1, . . . ,

Dn

THE FUNDAMENTAL THEOREM V IA GROTHENDIECK-CARTIER

a = ( « ! , - ■ , . . cc ) > nJ a at, x“

a l = a.T I 9, . ,. . - Of ] 9 n i a i = 2 a± , 00, > ° a QO Daf - D ,1 * .. . . • Dnn ■ ■ we can map

i>

ham om orphically in t o

ft -

k [[X j, . . . ,

^

D^f x“

Xn ] ]

v ia

= A(f)

0 < | a | < °° (where

bi s

the image o f

an elem ent

b € v

th e g e n e r a l th e o ry o f com plete l o c a l r in g s , B:

_

x^

in

k) . On the o th e r hand, by

th e re i s a s u r je c t io n

k [ [ X . , , . . . , X ^ J — - i)

g (mod m ) . Then

such

th a t B(X^) =

. . ,

Xq ] ] i n t o i t s e l f in d u c in g the i d e n t i t y modulo

fo r e

A o B

th a t

A

i s an automorphism;

A ° B

and sin c e

is B

a homomorphism is

of

( X - , , . . . , XQ) ~ . s u r je c tiv e ,

k[ [X 1, T h ere­

t h is im p lie s

i s an isom orphism . QED COROLLARY:

n o n - s in g u la r .

If

c h a r(k )

= 0 , then a l l the schemes

P (| )

are

T h e re fo re dim P (S ) = dinij^H1( F , op )

.

P r o o f : By C a r t i e r 's theorem , and the isomorphism ta n gen t space o f

P (t)

at

0

w ith

H1(F ,

o f the

Z a r is k i

O p ).

T h is p ro v e s E x is te n c e Theorem ( A ) , and r e -p r o v e s the theorem o f L e c tu re 23, f o r cu rves

D

such th a t

H1(F , O p(D )) = ( 0 ) .

LECTURE

26

RING SCHEMES; THE WITT SCHEME §0.

O u tlin e I n s e c t io n 1, th e v ie w p o in t o f the r in g schemes i s in tro d u c e d ,

w ith some b a s ic d e f i n i t i o n s and c o n s tr u c tio n s . I n s e c tio n 2, we d e v e lo p th e W it t r i n g scheme a s s o c ia te d w ith a prim e

p

and a p p ly i t

to the problem f o r w hich i s was o r i g i n a l l y used—

th e in v e r s io n o f a fu n c to r which one would n o t o ffh a n d have su spected was in v e r t ib le I

The problem i s d e v e lo p e d i n p a r ts

i s d e s c rib e d i n p a r t C, and i t

A

and

B, the W it t scheme

i s used t o s o lv e the problem in p a r t D.

The

re a d e r w is h in g t o sk ip t h is t a n g e n t ia l d is c u s s io n can rea d p a r t C o n ly . I n s e c tio n 3, p a r t A , we d e v e lo p the " u n iv e r s a l W it t schem e," a m o d ific a t io n o f the c o n s tr u c tio n o f §2;

(a " g e n e r a liz a t io n " i n th e sense

th a t the W it t scheme a s s o c ia te d w ith any prim e c a t in g " the u n iv e r s a l schem e).

We use i t

lo g a r ith m s "- a r i n g whose a d d it iv e tiv e

1.

can be g o tte n by "tr u n ­

s tru c tu re i s isom orp h ic t o the m u lt ip lic a ­

s tru c tu re o f the s e t o f fo rm a l power s e r ie s

f i r s t c o e ffic ie n t

p

in p a r t B t o o b ta in a " r i n g o f (o v e r a g iv e n r in g

R) w ith

In p a r ts C, D and E, we d e s c r ib e c e r t a in mappings

and tru n c a tio n s o f the W it t scheme, f o r w hich we s h a ll have use l a t e r in d e a lin g w ith power s e r ie s . §1.

G e n e r a lit ie s In any c a te g o r y

o b je c t

P, we

H an o b je c t ,

C h a vin g d i r e c t p ro d u c ts , and h a vin g a f i n a l

can d e fin e " r i n g o,

i,

o b je c t s " :

v , a, and

s e x tu p le s

(H , o,

i,

v , a,

n ),

\± maps:

o:

P

“ *■ H ( z e r o elem en t)

i:

P

—*■ H (u n it y )

v:

H

a:

H x H -*■ H (a d d it io n )

H (a d d i t i v e in v e r s e )

ii:

H x

H

H

(m u lt ip lic a t io n )

which s a t i s f y the ob viou s g e n e r a liz a t io n s o f the r i n g axioms f o r s e ts and s e t maps.

*

We s h a ll n ot count 1 ^ 0 r in g .

among th e r i n g axiom s; we a llo w th e t r i v i a l 1 71

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE G iven any o th er o b je c t s tru c tu re i s induced on from

C

A

o f our c a te g o r y , we fin d th a t a r in g

h H( A ) , so th a t

h H becomes a c o n tr a v a r ia n t fu n c to r

to R in g s . We are a c t u a lly a lr e a d y f a m i l i a r w ith some examples o f r in g ob­

j e c t s in the c a te g o r y o f schemes. non-commutative r in g scheme.

The v a r i e t y o f a l l

A sim p ler example i s

n x n

m a tric e s i s a

the a f f i n e l i n e , which

has an obviou s r in g scheme s tr u c tu r e . Though our d e f i n i t i o n s h old in the c a te g o r y Schemes^ over

an

a r b it r a r y scheme

over

Spec z

S, we s h a ll here

o f schemes

o n ly be w ork in g w ith r in g schem

(" a b s o lu t e r in g schemes") and c e r t a in lo c a l i z a t i o n s o f

Z.

A ls o , in a l l cases which we s h a ll d e a l w ith , the u n d e rly in g schemes w i l l be a ffin e .

The maps d e fin in g the r in g scheme s tru c tu re s w i l l thus be g iv e n by

r in g homomorphisms.

These w i l l go in the o p p o s ite d ir e c t io n t o the scheme

maps (s in c e the r e l a t i o n between a f f i n e

schemes and r in g s i s

c o n tr a v a r ia n t)

but th ey w i l l a c t u a lly be the ex p e c te d e q u a tio n s , look ed a t d i f f e r e n t l y . Thus, where we would be accustomed to d e s c r ib in g a d d itio n on the a f f i n e l i n e as the map (x , x 1) -► x " r in g term s, the map

(s e n d in g

Z [X ]

A1 x A1 —►A1)

Z[X]

0 z [X ]

(A le s s t r i v i a l example i s in g to each r in g

R

the r in g o f p a ir s Spec Z [X , Y ]

a (X )

= X 0 1 + 1

X

a (Y )

= Y 0 1 + 1 0 Y

(x , y )

h^ (Ct)

sake

of

eR , w ith

y ')

^i(X) = X 0 X - Y 0 Y n (Y ) = X 0 Y

(i

+ Y 0 X

.

( f o r the moment), t o what elem ent o f the

does the i d e n t i t y map corresp on d ?) We s h a ll here be in t e r e s t e d in r in g the a s s o c ia te d fu n c to rs

hH .

c la s s o f f u n c t o r ia l c o n s tru c tio n s o f r in g s tia lly

term wise a d d it io n ,

^ .(x x * - y y ', x y ' + x ’ y ' ) .

w ith a d d itio n

and m u lt ip lic a t io n

C a llin g t h is scheme r in g

X -*• X 0 1 + 1 0 X.

the "Argand plan e fu n c t o r ," a s s o c ia t -

and w ith m u lt ip lic a t io n g iv e n by (x , y ) ( x ' , I t i s re p re s e n te d by

x " = x + x 1, i t becomes, in

determ in ed by

schemes

The

H m a in ly f o r the

r in g schemesre p r e s e n t a c e r t a in

h u (R)

from r in g s

R.

(E ssen -

th ey g iv e th ose c o n s tru c tio n s in which the r e s u lt in g r in g can be d e ­

s c rib e d as the s e t o f a l l n -tu p le s

(n f i n i t e

or i n f i n i t e )

o f members o f

R

s a t i s f y i n g c e r t a in p o ly n o m ia l c o n d itio n s , and where a d d itio n and m u lt ip lic a ­ t io n are g iv e n by p o ly n o m ia l fu n c t io n s .) A r in g scheme over some l o c a l i z a t i o n o f

z

w i l l correspon d to a

c o n s tr u c tio n i n ,which the p oly n o m ia ls used may in v o lv e c e r t a in f r a c t i o n a l c o e f f i c i e n t s , and which thus can o n ly be a p p lie d to th ose r in g s in which c e r * t a in in t e g e r s are i n v e r t i b l e . One fu n c to r which i t i s easy to r e p r e s e n t i s th a t a s s o c ia t in g t o a r in g

R

the r in g

R [[X ]]

o f fo rm a l power s e r ie s in an in d e te rm in a te . We

s h a ll c a l l the r e p r e s e n tin g r in g scheme Spec Z [A Q, A 1, . . . ] e ffic ie n t s

(where

v .

The u n d e rly in g scheme i s

the A 's a re in d e te rm in a te s , r e p r e s e n tin g the c o ­

o f the power s e r i e s ) , and the a d d it iv e and m u l t i p l i c a t i v e maps

a re g iv e n ( i n

terms o f the r in g

Z[A Q, A 1, . . . ] )

by

RING SCHEMES: THE WITT SCHEME

1 73

oc(A±) = A± ® 1 + 1 ® A^_ i

XAJ

= X Aj ® Ai - j J=0

The tru n c a te d power s e r ie s r in g s , th e (f in it e - d im e n s io n a l) schemes o f in t e g e r s

V.

schemes

R tX l/X 11,

vn = Spec z [A Q, . . . ,

These form a p r o j e c t i v e system :

m < n

'

fo r every

th e re i s a tru n c a tio n map from

to the in c lu s io n :

Z [A Q, . . . ,

C Z [A Q, . . . ,

a re re p re s e n te d by

A ^ .,], to

An-1 ] ,

q u o t ie n t - r in g — p a ir o f p o s i t i v e

Vm and

corresp o n d in g v

is

the i n ­

v e r s e l i m i t o f t h is system . ( Some random n o te s on r e p r e s e n t a b i l i t y o f fu n c to r s o f R in gs -*■ R in gs by r i n g schemes: Such fu n c to r s must have the p r o p e r ty h(R ® R ') = h (R ) h ( R ') , hence the fu n c to r sen din g e v e r y r in g t o a f i x e d r in g A cannot be r e p r e s e n te d . (But one can c o n s tru c t a scheme w hich sends e v e r y r in g w ith connected spectrum to Z — i t i s a d is c r e t e union o f c o p ie s o f Spec Z. If A i s i n f i n i t e , t h is i s no” a f f i n e , s in c e i t i s n o t compact. If A -+ B i s a 1-1 map o f r in g s , h (A ) -*■ h(B ) must be a 1 - 1 map o f r in g s . Hence th e fu n c to r R -*■ R/p cannot be r e p r e s e n te d : the 1-1 map z -*■ Q g iv e s

z /p —♦* 0 .

Though the ’’power s e r ie s r i n g ” fu n c to r can be r e p r e s e n te d , th e ( f i n i t e ) ’’p o ly n o m ia l r i n g ” fu n c to r a p p a re n tly c a n 't . What would be a "g e n e r ic f i n i t e n o ly n o m ia l? " I )

§2 .

p -a d ic r in g s and th e W it t fu n c to r Most o f t h is m a t e r ia l appears in S e r r e , Corps Locaux, but the

p r e s e n ta tio n th e re i s more r a p id , and i t

i s done somewhat d i f f e r e n t l y :

the

form a lism o f r i n g schemes i s n o t th e re used.

A: M u s ic a l C h airs ( w h ile s h rin k in g ) Let

p

Let

A

own r a d i c a l ( i . e . ,

be a prim e number. be any r i n g i n w hich

such th a t

A/p

a re in d i s t i n c t re s id u e c la s s e s (a - b )^ 4 0 (mod p) .

p

g e n e ra te s an i d e a l w hich i s i t s

has no n i l p o t e n t s ) . mod p ,

I n o th e r w ords:

Then i f

two elem ents

so a re t h e i r p - t h pow ers:

a^ - b^ =

the F roben iu s endomorphism o f

A/p

is

1 -1 . However, i f

two elem en ts a re the same c la s s mod p , t h e i r p - t h

powers w i l l be i n the same c la s s (a + p x )p =

2

mod p :

aP + (p )a p " 1px + ( ^ ) a p _ 2 ( p x ) 2 + . . . = aP (mod p 2)

More g e n e r a lly , r e p la c in g s ee:

i f two elem en ts a re congruent

w i l l be congruent be congruent

mod pk+1, k+n

mod p

a + px

by

modpk (k / 0 ) ,

a + p^x

i n the ab ove, we

then t h e i r p - t h powers

whence, by in d u c tio n , t h e i r pn- t h

powers w i l l

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

m

Thus th e o p e ra tio n o f r a i s i n g to the p - t h pow er, though i t the congruence c la s s e s

(mod p)

under the p -a d ic m e tr ic .

d is tin c t,

Sin ce the Froben iu s endomorphism o f

n o t, i n g e n e r a l, be the i d e n t i t y ,

keeps

causes each t o ’’ sh rin k down” A/p

w ill

th ese congruence c la s s e s w i l l be p la y in g

a w ild game o f m u sica l c h a irs as th ey sh rin k , c o n fu s in g the s it u a t io n a b i t . However, suppose th a t iu s endomorphism i s of

A.

F or e v e r y

in

[a ].

o th e r [a ]. in

1-1 o n t o .) n,

(I.e .,

th a t the Froben­

be any congruence c la s s

(mod p)

(p n) - t h power

(p n) - t h powers ol* i t s members a re a l l congruent t o each

we g e t a c a n o n ic a l congruence c la s s

Furtherm ore, as [a ]

is p e r fe c t.

[a ]

some congruence c la s s w i l l have i t s

Sin ce the

mod pn+1,

A/p

Let

n

mod pn+1

d e fin e d in

in c r e a s e s , each s u c c e s s iv e su b -co n gru en ce-cla ss

w i l l b e lo n g t o the p re c e d in g . C le a r ly , what i s b e in g d e fin e d i s a member o f

co m p letio n o f

A.

make t h is m e a n in g fu l.) LEMMA 1 . such th a t

A/p

A , the p -a d ic

(We should here assume the p - a d ic to p o lo g y se p a ra te d , to Or, assuming Let

A

is p e r fe c t.

A

com plete t o b e g in w ith , we g e t :

be a r in g com plete under the p -a d ic to p o lo g y , Then th e re i s a c a n o n ic a l map

sen din g each re s id u e c la s s t o i t s unique member w hich has fo r a l l

n.

phism o f

f

can be c h a r a c te r iz e d

A / p -► A

which

f :

A/p

-► A

(p n) - t h r o o t s

as the unique m u l t i p l i c a t i v e homomor­

s e c tio n s the q u o tie n t map A-** A/p.

(P r o o f o f

the

l a s t sentence l e f t to the r e a d e r .) E xam ple:

If

A

is

sim ply the r in g o f p -a d ic in t e g e r s

f (A/p)

c o n s is ts o f

th e ( p - l ) - s t r o o t s o f u n ity and z e r o .

B:

The T e ic h m u lle r c o n s tr u c tio n I t i s w ell-kn ow n th a t a p - a d ic number can be re p re s e n te d u n iq u e ly

by a ’’power s e r i e s ” t h is i s

of little

0, 1 , . . . ,

p -i

aQ + a ^

+ a 2p

+ ...

where

a^ = o ,

1 ,...,

p -i .

But

m ath em atical i n t e r e s t , because the s e t o f r e p r e s e n t a t iv e s

o f the r e s id u e c la s s e s

mod p

is

c l e a r l y r a th e r a r b it r a r y .

Now, how ever, we have a b e a u t if u l f u n c t o r i a l s e t o f r e p r e s e n ta ­ t i v e s o f the re s id u e c la s s e s .’ r in g s

A

Making use o f them (and g e n e r a liz in g t o the

d e a lt w ith i n the p re v io u s s e c t io n — we need o n ly add the hypo­

t h e s is th a t

p

n ot be a

z e ro d i v i s o r i n

A, so th a t th ese power s e r ie s w i l l

be u n iq u e ), we g e t : LEMMA 2 ; z e r o - d iv is o r ,

betw een members o f g iv e n by

Let

such th a t A

A A/p

be a com plete p - a d ic r i n g where is p e r fe c t.

and sequences

p

i s not a

Then th e re i s a 1-1 correspondence

(| Q, |. , , . . . )

o f elem en ts o f

A/p,

RING SCHEMES: THE WITT SCHEME

175

Suppose we can d is c o v e r how t o c a lc u la t e i n s e q u e n c e -re p re s e n ta tio n s . the s tru c tu re o f

A

Then i t

should f o l l o w

from th a t o f

A

u s in g th ese

th a t we can r e c o n s tr u c t

A/pJ

I t w i l l tu rn out th a t we can do t h i s , but the r e s u lt s w i l l be in a more co n v e n ie n t form i f we u se,

n o t p r e c i s e l y the above corresp on d en ce,

but th e correspondence t 2, . . 0

«-

f ( t 0 ) + p f ( e ? 1) + p 2 f (?2 2)+ •••

(1)

d 0, t ,,

(T h is can be

seen from th e example worked out in Appendix A .)

C:

The W it t Scheme (an apparen t in t e r lu d e )

in to

A'00 = Spec Z[WQ, . . . ]

Let

w

be th e scheme Spec Z[XQ,

(2 )

•

and l e t us map

w

by th e map g iv e n by th e W it t p o ly n o m ia ls :

WQ = XQ V,

= x P £+ pXl

W2 = X0 + PX? + P 2x2

Wn = xPn + pXPn (p

is

s t i l l a f i x e d p rim e.

the c o o rd in a te s o f

A°°,

+ ...

+

Note th e c o n fu s in g te rm in o lo g y :

and th e

X *s

the c o o rd in a te s o f

D e fin e a r i n g scheme s tru c tu re on

A**

the

W, s

a re

w .)

by the maps

a(W ) = Ws 0 1 + 1 ® Wg a ll n(Ws ) =

A00 r e p r e s e n ts th e fu n c to r th a t a s s ig n s to the r in g sequences

(wQ, w ^ . . . )

m u l t ip lic a t io n , i . e . ,

o f elem en ts o f

R

R, the r in g o f i n f i n i t e

under componentwise a d d it io n and

the d i r e c t p rod u ct o f i n f i n i t e l y many c o p ie s o f

We c la im th a t th e r i n g scheme s tru c tu re on scheme s tru c tu re on

s.

ws X ' ) )

+ term s in v o lv in g lo w er cp's

=

--------------------------------- ---------------------Pk

*

As b e fo r e , i f

p/n,

we s e t

W /n = 0.

Note th a t by ” lo w er X* s , " we

mean, o f co u rse, X*s whose in d ic e s a re p ro p er d iv is o r s o f

n.

RING SCHEMES: THE WITT SCHEME S in ce the "lo w e r cp's" a re " i n t e g r a l " d i v i s i b l e by

p) by in d u c t iv e h y p o th e s is , i t

(i.e .,

have no denom inators

s u f f i c e s to show

»(w n ( X ) , ¥n ( I ' ) )

I 1) ) (mod pk)

T h is we do e x a c t ly as b e fo r e : pk , and "commute" 0

and

.

we s u b s titu te our "wn (X )

mula in the le ft - h a n d s id e , now d is c a r d in g the " t a i l " v i s i b l e by

181

wn /p>

=" f o r ­

term s in c e i t

is d i­

so th a t the d e s ir e d congruence

becomes wn /p (cp(Xp , X 'p ) )

s wn /p (q>p (X , X ' ) )

(mod pk)

.

T h is h o ld s by our sub lemma.

gum

Hence a l l the c o e f f i c i e n t s must l i e

in

So, as b e fo r e , we g e t a r in g scheme

Spec Z [X ,,

w ith a homomorphism

Spec ZIW ,, W2, . . . ]

w hich becomes an Isom orphism on t e n s o r ln g w ith

B:

w,

z .

Spec ( Q ) .

Logarithm s o f power s e r ie s R e c a llin g th a t

d e s ig n a te s the " fo r m a l- p o w e r - s e r ie s " r in g

V

scheme, l e t us d e s ig n a te by e q u a tio n

AQ = 1.

V°

the c lo s e d sub scheme co rresp o n d in g t o the

T h is r e p r e s e n ts power s e r ie s w ith co n sta n t term

1, and

i s a commutative group scheme under th e r e s t r i c t i o n o f m u lt ip lic a t io n in o We s h a ll w r it e the R -va lu ed p o in t (1 , a 1, a p, . . . ) of V i n the more f a m i l i a r form its

1 + a .jt + a gt

fu n c to r o f

2

+ ...

.

We s h a ll d e a l w ith

v°

v.

i n terms o f

R -va lu ed p o in ts i n o rd e r t o make a v a ila b le t o us w ell-know n

r e s u lt s about fo rm a l power s e r ie s . C on sid er the f o llo w in g maps o f schemes: v w_ W x Spec (Q) — > ^ > -»

A00 x Spec (q)

where ♦ (w ,, V2, . . . )

= exp|^- ^

■►v° x Spec (Q) w m

1 tm

We cla im th a t the co m p o sitio n exten d s t o an isom orphism o f the schemes and

V °.

To check t h i s , we f i r s t recom pute t h is map on R -va lu ed p o in t s ,

in the case

R J Q.

Say, X

ai t;L = '*r( wi>

= * 0 w (x t , x 2, . . . )

.

W

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE We g e t :

2 V - » p [-£ “

E n xn

«"] m

n

d

a

= exp [ E log (1 ■

]

n

00

- n o n-1 The

and th e

x^

v

n>

•

a re now c l e a r l y m u tu a lly r e la t e d b y p o ly n o m ia l equa­

t io n s w ith i n t e g r a l c o e f f i c i e n t s . QED Now the map from

A°° x Spec Q

to

V° x Spec Q

i s a homomor­

phism from the a d d it iv e group s tru c tu re o f th e form er t o th e ( m u lt ip lic a ­ tiv e )

group s tru c tu re o f the l a t t e r .

homomorphism.

Hence th e com posite map i s

Hence th e scheme-isomorphism betw een

group homomorphism on a dense su b set, i s ,

w

and

such a

V °,

b e in g a

in f a c t , an isom orphism o f group

schemes: W i s a r i n g scheme whose a d d it iv e s tru c tu re i s (The q u e s tio n " t o what o p e r a tio n on

w

s tru c tu re o f

C:

th a t o f th e group scheme \_°. v°

does th e m u l t i p l i c a t i v e

corresp on d ?" i s in v e s t ig a t e d in Appendix B .)

T ru n ca tion s We can "tr u n c a te " th e p o w e r -s e r ie s rin g-scn em e

v

because i t s

a r ith m e tic o p e ra tio n s a re such th a t the n -th term o f the sum or p rod u ct d e ­ pends o n ly on the n -th and lo w er term s o f th e elem en ts g iv e n . n -th term depends o n ly on th ose terms whose in d ic e s d iv id e is ,

th a t g iv e n any s e t

o f a number i f

it

g e rs "tr u n c a tio n s e t s . " T^:

w:

W -*• A00

of

W.

we s h a ll c a l l such s e ts

For any tru n c a tio n s e t

w, th e

The r e s u lt ws = S

o f in t e ­

S, we g e t a tru n c a tio n

W -► Wg.

V a rio u s f a c t s a re map

In

S o f p o s i t i v e in t e g e r s w hich c o n ta in s e v e r y d i v i s o r

co n ta in s th a t number, we g e t a r in g scheme

Spec z [ x s ] s€s> a " tr u n c a tio n " homomorphism

n.

tru n c a te s t o

t r i v i a l t o v e r i f y about th ese schemes: a map

wg :

Wg -♦ AS,

The

and th e r i n g s tru c tu re

RING SCHEMES: THE WITT SCHEME on

Wg

is

th e unique s tr u c tu r e making t h is a r i n g homomorphism.

tru n c a tio n s e ts TS S ' o

Wr 1 ^

S C S’ ,

we g e t a tr u n c a tio n homomorphism

G iven two

Tg g , : Wg,

° TS' S" = TS S" * W i t s e l f i s > co u rse, wz+ , and i is Wp , the scheme c o n s tru c te d in §2. The scheme

f •••J

q

W|-1 n -1 } are i somorP h ic to the tru n c a te d power s e r ie s groups Vn , hut the o th e r tru n c a tio n s do n o t corresp on d to any f a m i l i a r c o n s tr u c tio n w ith power s e r ie s r in g s . We neea some g e n e r a l nonsense a t t h is p o in t : f:

maps o f grou ps: if

the

hf (X )

hf (X ) :

h^(X)

-*■ h g(X ) a re 1-1 f o r

A homomorphism

" 1- 1" i f

A -*■ B o f commutative group schemes w i l l he c a lle d

X,

"o n to ”

a re a l l on to.

The p r o p e r ty o f b e in g 1-1 behaves q u it e n i c e l y . le n t , by d e f i n i t i o n ,

whose fu n c to r i s

(We c o n s tru c t

K

I t i s e q u iv a ­

to b e in g a monomorphism in the c a te g o r y o f schemes.

G iven an a r b it r a r y homomorphism K A

the induced

a l l schemes

f:

A -► B,

we can g e t a 1-1 homomorphism

the fu n c to r o f k e rn e ls o f the induced group maps.

as the f i b r e i n

A

o f the Z -va lu ed p o in t

do we show th a t th e group o p e r a tio n l i f t s

to

0

of

B.

How

K ?)

On the o th e r hand, th e p r o p e r ty we have c a lle d b e in g "o n to " i s s tro n g e r than b e in g an epimorphism b o th o f schemes and o f group schemes. I t i s e q u iv a le n t t o the e x is te n c e o f a scheme map w hich

" s e c t io n s " f — a r i g h t in v e r s e map.

see th a t i t

g

T h is i s

from

B

back t o

A

c le a r ly s u ffic ie n t ;

to

i s n e c e s s a ry , we n o te th a t by our d e f i n i t i o n o f " o n t o " , the

i d e n t i t y map i n

h g(B )

id e n tity .

g

(But

must come from a map

g

in

h^(B)

such th a t

fg =

w i l l n o t i n g e n e r a l be a group scheme homomorphism.1)

We cannot in g e n e r a l c o n s tru c t a group scheme w ith the p r o p e r ­ t ie s

o f a c o k e rn e l o f

f.

Hence, though e x a c t sequences can be d e fin e d (b y

the c o n d itio n th a t the induced sequences

-► h^(X )

h g(X )

-»■ h^(X)

-►

a l l be e x a c t-n o te th a t t h is im p lie s th a t the k e r n e l o f each map i s a c o ­ k e r n e l to the p r e c e d in g ), th e y a re n o t so ea sy t o come b y. an onto map

A

B

0,

we can g e t an e x a c t sequence

N ote th a t the c o n d itio n s " 1 - 1 , "

However, g iv e n

0 -► Ker f

A

B

0

"o n to " and " e x a c t " r e s p e c t

base e x te n s io n . The tru n c a tio n maps we have d e fin e d a re a l l on to: we g e t a s e c t io n X i

D:

Wg

back to

in any way we l i k e ,

Wg,

G iven

S C S’

by " f i l l i n g in " the m is s in g c o o rd in a te s

e . g . , w ith z e r o e s .

Ca n o n ic a l maps There a re two s e ts o f maps from

W

to

w

w hich a re u s e fu l.

1 8l*

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE a)

D e fin e

Vn :

» — W

by

V n 0

lf

n |m

o th e rw is e .

( I n terms o f R -va lu ed p o in t s , f o r in s ta n c e ,

(0, 0, x 1 , 0, 0, x 2 , . . . ) * ) i )'

x 2, . . . )

=

We c la im :

Vn o v m = Vnm

ii)

Vn

i s an a d d it iv e isom orphism from

o f the tru n c a tio n i)

is

o b vio u s, and one checks th a t

phism o f the scheme

w

onto the k e r n e l

Vn

i s a t l e a s t an iso m o r­

w ith t h is k e r n e l by lo o k in g a t R -va lu ed p o in t s .

check th e a d d itiv e n e s s , i t induced map on

w

Tz+ _n z+

s u f f i c e s t o te n s o r w ith

A00 x Spec q

i s a d d it iv e .

To

and show th a t the

q

We f in d , in f a c t ,

th a t i t

is

d e s c rib e d by W

H*

■.f nWm/n if n|m L 0 o th e rw is e . QJED

F or any tru n c a tio n s e t

S, we ob serve th a t we have, s i m i l a r l y , a map VS ,n :

where

S/n = {me z|nm e S),

wS/n

“ *■ WS W5 /n

w hich i d e n t i f i e s

w ith th e k e r n e l o f

the t r u c a t io n

WS b)

D e fin e

isom orp h ic scheme e ffic ie n t

1.

wS-nz+

v°:

le t

P (t )

W -+ w

Fn :

by i t s a c t io n on R -va lu ed p o in ts o

be a power s e r ie s in

L e t us d e s ig n a te by

Tn

t

w ith f i r s t c o ­

the fo rm a l n -th r o o t s o f

t;

then the p rod u ct n p ( Ti ) i b e in g symmetric i n the

t

> s ,

w i l l a g a in be a power s e r ie s in

c o e f f i c i e n t s w i l l be p o ly n o m ia ls in the c o e f f i c i e n t s o f o f the map d e fin in g the r e l a t i o n between

v°

and

A00

P.

t,

and i t s

An exam in ation

shows us th a t

Fn

correspon ds t o th e map (W-| ) W2, . . . ) o f R -v a lu ed p o in ts o f Fn

A00.

i s a r i n g homomorphism.

-

(wn , w2n> . . . )

We n ote th a t t h is i s a r i n g homomorphism, so A ls o

Fn ° Fm = F ^

.

185

RING SCHEMES: THE WITT SCHEME

We deduce (b y the u su a l Mo n ly - t h o s e - in d ic e s - t h a t - d e v id e - m M a r ­ guments) th a t s im ila r maps ( a l s o r i n g homomorphism) a re d e fin e d betw een the tru n ca ted schemes: P S ,n : c)

Look a t

WS

Fn ° Vn :

wS/n

•

ch eck in g i t

on R -v a lu e d p o in ts o f

we f in d :

-if n#

Fn o Vn = m u lt ip lic a t io n by I n some c a s e s ,

one can d iv id e by

LEMMA 5: P r o o f:

n

n:

i s i n v e r t i b l e in

W x Spec Z [ l/ n ],

We r e c a l l th a t one can tak e n -th r o o t s o f monic power

s e r ie s i f we a llo w d i v i s i o n o f th e c o e f f i c i e n t s by by

n

in

A00,

n;

hence one can d iv id e

w x Spec z [ i / n ] . QED Thus, o v e r Spec Z [ l / n ] ,

E:

Vn

i s a r i g h t in v e r s e to

D ir e c t p rod u ct d eco m p osition s The d i r e c t p rod u ct o f two r i n g schemes

w ith th e te n s o r p r o d u c t!) the schemes f o r

H

and

(Do

in

G :

G

th e elem en t ( 1 , 0 )

=

H

x

of

e

A00 = Spec z[W1, W2, . . . ]

I o f th e p o s i t i v e in t e g e r s ,

^

and

H '

S

is

S of

=

0

S -va lu ed idem potent s , and

H

and

H 1

are

r e s p e c tiv e ly . o v e r Spec z .

A00 has a ( Z -va lu ed ) * rij(Wi ) = 1

th e re i s a 1-1 c o r ­

G ,

i s an

H x H *

the k e r n e ls o f m u lt ip lic a t io n by 1-e and Look a t

ov e r

I t s u n d e r ly in g scheme i s th e p rod u ct o v e r

S t a r t in g w ith a commutative r i n g scheme e

H !

n o t confu se t h is

H 1 .

respondence betw een d eco m p o sitio n s p o in ts

and

H

d e fin e d ju s t l i k e the d i r e c t p rod u ct o f two r in g s .

set

Fn .

i

€ I

i

4 I

F or e v e r y sub­

idem poten t p o in t

and c o r r e s p o n d in g ly decomposes:

We mean, o f c o u rse, the rin g-sch em e o p e r a tio n o f m u lt ip lic a t io n by n, which does n o t correspon d t o c o o r d in a te -w is e m u lt ip lic a t io n by n e x c e p t f o r th e c o o rd in a te s w^ o f A00. The same should be u nderstood in the f o l ­ lo w in g lemma, co n cern in g m u l t ip lic a t io n by th e Spec z [ 1 /n] -v a lu e d p o in t ” i /n.

186

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE Hence

adm its a l l th ese d ecom p osition s t o o , o v e r

W

The q u e s tio n a r is e s :

suppose

P

Spec Q.

i s any s e t o f p rim es, and

9 = Spec z [ . . . , 1 / p ,. . . ] p^p Then how many o f th ese d ecom p osition s o f

9 ?

E q u iv a le n t ly , which o f the

no prim es

in

P

can be asked f o r su bsets Let

Q,

w by a tru n c a tio n

be the s e t o f prim es n o t in

d e s ig n a te the m u l t i p l i c a t i v e

by

P

(r e s p e c t iv e ly

Q)

P r o o f: L1£j

+

enZ +

is

the same ques­

and

P.

Let

P( r e s p e c t i v e l y

semigroup o f p o s i t i v e in t e g e r s gen era ted

1.

Note th a t th e s e ts

nP

fo r

n € Q

Z+ . LEMMA 6:

s

9 — have

ICS.

Q)

p a r t it io n

a c t u a lly occur over

are r a t i o n a l o v e r

o c c u rrin g i n the denom inators o f t h e ir co o rd in a te s ?

C le a r ly , i f we r e p la c e t io n s

w 0 Spec Q

= w- 1 ( t ^ )

sim p ly

I t s e lf Is

For any For any

I V

11

1.1

n e Q, n e Q,

9.

we n ote th a t the p r o je c t io n g iv e n by

F , hence i s

r a tio n a l over

11

r a tio n a l.

9;

i n p a r t ic u la r ,

Now

EnP

T h is i s ,

en— e W i s r a t i o n a l o ver

=

( snZ *

” ep n z + )

fo r m a lly , an i n f i n i t e p rod u ct.

However, i t

"c o n v e rg e s "

c o o rd in a te w is e in the sense th a t each c o o rd in a te i s co n sta n t a f t e r a c e r t a in number o f term s.

T h is i s

c le a r in the

tru e in the w - c o o r d in a t e s .

COROLLARY: i s r a t i o n a l o ver

A°° c o o r d in a te s , hence i t

i s a ls o

Hence the le ft - h a n d term i s r a t io n a l.

For any tru n c a tio n s e t

S

and

n € Q,

enpns €

9.

LEMMA 7: L e t

S

X_

= b P r o o f:

If

p r o o f would be c le a r . w ith o u t f i n i t e n e s s .

neQ

be a tr u n c a tio n s e t . e„^-nQ( ^ b

wQ)

Then o ver

( a l l schemes ten s o re d w ith

9).

s

our s e t o f idem poten ts were f i n i t e , It

9

the method o f

turns out th a t we can here a p p ly the same p r o o f

We are t o v e r i f y th e u n iv e r s a l p r o p e r ty o f p rod u cts

on X -va lu ed p o in t s . G iven a fa m ily o f maps an : X -► enpng ( ws ^ 9 we take the map ocn : X Wg. T h is i n f i n i t e sum i s d e fin e d by e x a c t ly the same re a s o n in g used in the

l a s t lemma, and i s c l e a r l y the unique mapwhose

com p osition s w ith the p r o je c t io n s g iv e the

a .

187

RING SCHEMES: THE WITT SCHEME

tain

COROLLARY: n € Q, then LEMMA. 8:

If a set I is the union of sets is rational over 9 . Let n e Q,

enPhs^ WS^ Proof:

and “

S

for cer­

be any truncation set.

Wpns/n

Then

^a11 scIieines tensored with

9) •

Consider the maps

,

PS,n

enpn s < V

nP n S

projection

WS .

truncation

wS/n

.

n ,. u/n;vS,n

»p n s/n

any section of truncation

All are rational over 9 . It will suffice to show that the com­ position of the maps going to the right and the composition of the maps go­ ing to the left are ring scheme homomorphisms, and are inverses to one an­ other. Tensoring with Spec Q and using the A-coordinates, we verify easily that this is so.

QED Hence we have, for every truncation set primes,

S

and set of

P:

en p n s ( * s ® ? )

*

"sns/n®

?

(One might want to know whether what we have achieved is al­ ways a maximal direct product decomposition of Wg 9; equivalently, whether the for I a union of sets if fl S (n e Q) are the onl^ idempotents of wg. We prove in Appendix C that this is so.)

APPENDICES TO LECTURE 26

A ).

(C f. end o f §2B, p.

175)

We s h a ll f ig u r e out e x p l i c i t l y how to add the f i r s t two terms o f s e r ie s fo r

o f th e typ e o r i g i n a l l y p roposed ( P 0) ) *

sQand (f(a )

+ p f(b ))

+ ( f ( a ')

R educing mod p , due c la s s

Whatwe must do

is

s o lv e ,

s 1, th e congru ence:

a,

we g e t

+ p f ( b ') )

= f ( s 0) + p f ( s , )

and r e c a l l i n g th a t

f(a )

(mod p 2) .

b e lo n g s t o th e r e s i ­

a + a* = sQ.

S u b s titu tin g t h is back in , and i s o l a t i n g the term in

s 1,

we g e t p f ( s 1) = p f ( b )

+ p f ( b ')

+ ( f(a )

+ f ( a 1) - f ( a + a 1) )

(mod p 2) .

We know th a t th e l a s t e x p r e s s io n i s a m u lt ip le o f cou ld e x p ress i t fin is h e d .

The problem i s

i s as f o l l o w s : a + a’,

as such, we cou ld " d iv id e th rou gh " t o g e t an e x p r e s s io n f o r

( f ( a 1/p ) + f ( a ,1 /p ) ) p

and, b e in g a

by

p

p.

I f we

and would be

f(a + a ! ) .

The s o lu tio n

b e lo n g s t o the congruence c la s s

p - t h pow er, must b e lo n g t o the su b class

mod p

of

f ( a + a 1) ! Now a p o ly n o m ia l i n

(x + y )p

can be w r it t e n

x

y

and

( f ( a 1/p ) + f ( a ' 1/p ) ) p

ee

f(a )

p f ( s 1) = p f ( b ) f ( s 1) = f ( b )

+ f ( a ')

+ p t fC a 1^ ) ,

+ p f ( b ' ) - p [ f ( a 1/p ) , + f ( b 1) - [ f ( a 1/p ) ,

s1 = b + b ' So

xp+ yp+ p [x , y ] , where

w ith i n t e g r a l c o e f f i c i e n t s .

Hence

[x , y ] i s f(a + a ! ) =

f ( a ,1 /p ) ] .

Hence

f ( a ,1 ^p ) ]

(mod p 2)

f ( a ,1//p) ]

(mod p)

- [ a 1/p , a ,1 ^p ]

(a , b , . . . ) + ( a ’ , b 1, . . . )

= (a + a T, b + b l - [ a 1^p , a ’ 1^p ] , . . . )

I f we would l i k e a s e t o f c o o rd in a te s in w hich we can compute p u r e ly by p o ly n o m ia l o p e r a tio n s , we should e it h e r s u b s titu te s u b s titu te

b = p1^p .

a = o;p

or

The f i r s t c h o ic e would be unw ise, sin c e when we

b r in g in the t h ir d term o f the exp an sion , we would have t o change a g a in , and so on.

The second c h o ic e i s

the exp an sion ( 1 ) , (a ,

0 ,...)

the one we made in the t e x t .

In terms o f

the above r e s u l t i s : + (a ’ , 0’ , . . . )

= (a + a 1, 0+0’ - [a , a 1] , . . . ) 189

.

.

190

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

B) .

(Cf. end of §3b, p. 182) We want to investigate the "multiplication" induced on V° by the isomorphism with w. We shall, as usual, look at R-valued points. What we have to describe is then a binary operation on power series, which we shall write "°". We find first of all, using the formula for the isomorphism between A°° x Spec Q and v° x Spec Q that (1-at)0(1-bt) = 1 - (ab)t, when a and b are members of any ring containing q. It follows that this must hold for a and b in any ring. Since o distributes with respect to multiplication, we get m n m,n

n 0 - “j_t)»n (1- Pjt) = n (1 -

•

For the sake of simplicity, let letus us call call the the a (rather a than (rather than the 1/a.) /a^) the "roots” "roots" of nm (1 - a.t) a^t) . (Under (Under this this definition, definition, aa polynomialhas hasan anindefinite indefinitenumber number of of zero zeroroots.) roots.) Over Overan an algebraically algebraically closed field k, then, we can describe °° precisely precisely for for thethe finite finite (i.e., (i.e., terminating) power series: it is the thefunction function sending sendingany pair any of poly­ pair ofpoly­ nomials to the polynomial whose roots are areall all the thepairwise pairwise products products of of those of the given two. It is easy to see from this that the rational functions (quotients of polynomials) form a subring, which has, in fact, the structure of the "group ring" on the group k.* The full power series ring is the completion of this ring under a metric that makes two points "close" if the first n symmetric functions on them agree (though, of course, it takes some rigging to define the "symmetric functions" on a family some of whose members occur with nega­ tive multiplicity). This interpretation goes through in a more or less formal way for any ring without zero divisors. We can construct over any such ring a unique polynomial whose roots are all the pairwise products of the roots of two given polynomials, even if these roots don’t lie in the ring itself. The rational functions in the monic-formal-power-series group form a sub­ ring which can be thought of as the "semigroup ring" on the nonzero elements— It is amusing to note that a somewhat similar construction turns up in algebraic topology. A complex vector bundle on a space X induces a "Chern class polynomial over the ring Heven(X). It turns out that the operation "©" on bundles corresponds to multiplication of polynomials, while the taking of tensor products of bundles corresponds to the opera­ tion associating to a pair of polynomials the polynomial whose roots are all the pairwise sums of the roots ("in" H )of the given two.' Such an operation cannot be "defined in our power-series context, because the "in­ definite number of zero roots," which can be ignored under our "multipli­ cative multiplication," wreaks havoc with an attempt to set up an "addi­ tive multiplication." The essence of the problem is that our polynomials are of indefinite degree in t, while the topologist's polynomials have a definite degree, corresponding to the dimension of the bundle.

191

APPENDICES TO LECTURE 26

but we now must a llo w n o t o n ly fo rm a l sums o f elem en ts a c t u a lly in the r in g , but a ls o sums o f elem en ts ( i n t e g r a l l y )

a lg e b r a ic o v e r th e r in g ,

th ey appear i n f u l l s e ts o f c o n ju g a te s .

so lo n g as

The f u l l r in g i s a g a in a com ple­

t io n . The w

— the c o o rd in a te s o f the image in

A00 — a re the moments

Z o f. C ).

We s h a ll sk etch a p r o o f th a t the d i r e c t p rod u ct decompo­

Wg 0 9

s itio n o f

g iv e n in our f i n a l theorem i s maximal.

We f i r s t n o te th a t e v e r y idem poten t o f wQ o ver 9 S A 9 and the o n ly idem poten ts o f the l a t t e r a re

g iv e s an

idem poten t o f subsets the

I

e-j-.

o f S;

hence th e o n ly

p o s s ib le idem p oten ts i n th e form er are

What we d e s ir e t o show then i s

and o n ly i f ment i s :

I

i s a union

fo r every

p €

fo r

th a t

o f s e ts nP n S Pand elem en ts

m,

9

i s r a t i o n a l o ver

(n € Q ) . pm

An e q u iv a le n t

e S,

we have

if

s ta te ­

m € I

< = >

pm e I . It

c l e a r l y s u f f i c e s to

check t h is i n th e case P = ( p ) ,

s in g le t o n .

So suppose we had a r a t i o n a l

c o n d itio n .

Then th e re would e x i s t

th a t

m, p m , . . . , pk_1m e I ,

m e Q

p^Sn e S - I

n e c e s s a r y ).

C on sider the f a c t o r o f

co n ven ien ce)

co rre s p o n d in g t o

a tr u n c a tio n o f w hich i s

w ith

mP n S. k-,

k

g r e a t e r than z e ro such I

and

S -I

(we s h a ll drop the " 5 »s " I t w i l l be isom orp h ic t o .

I f we now f o l l o w

• ;P * through a l l th ese tr a n s fo r m a tio n s , we fin d th a t i t

prod u ct d eco m p o sitio n o f t h is scheme from w hich i t

a

n ot s a t i s f y i n g t h is

(in te r c h a n g in g

Wg

Wr-

and

I

if fo r

Wpn gy-m,

our idem potent

g iv e s us a d i r e c t

can be deduced th a t the

tr u n c a tio n : Wr 1

s p lits .

Wr ^ k -1 1 U ,...,p 1

^ki j

But i f wetake z/ p -va lu ed

p o in t s ,

( a l l schemes ten sored w ith

t h is means by the r e s u lt s

§2D th a t: z/pk

z/pk_1

s p lits .

C o n tr a d ic tio n !

of

LECTURE 27 THE FUNDAMENTAL THEOREM IN CHARACTERISTIC

a l l schemes

1° .

Let

H

X

over

k, k:,

be any r i n g scheme o v e r th e f i e l d H

d e fin e s a sh ea f o f rr.in g s

r (U , < H > ) In p a r t ic u la r , i f

A1

is

g iv e n i t s < A

i.e .,

p.

k.

< H >x

Then, f o r on

X v ia

= Hem, (U , H ) c a n o n ic a l r i n g scheme s tr u c tu r e , then

>x

-

°x

we r e c o v e r th e s tr u c tu r e sh ea f on

the c h a r a c t e r is t ic i s

p

>

X.

On the o th e r hand, suppose

Then u s in g th e W it t rin g-sch em e f o r

p, we can

g e t an in t e r e s t i n g s h ea f o f r in g s , B«°,x

=


:x n,

•

we g e t a sh ea f o f r in g s from th e tru n ca ted

scheme: ®n,X = < W( i , p , p 2, . . . , p n _1 ) x Spec k These sheaves o f r in g s form a p r o j e c t i v e

system o f sh eaves,

under th e ob viou s tru n c a tio n s Tn , n ,: w ith in v e r s e l i m i t =

o^ .

x,

®n,X

® n ',X

and w ith f i r s t term

(n > n T) ffl1 x

=

< A1 >x

These sheaves^were in tro d u c e d by S erre a t^ th e M exico C onference

in T o p o lo g y ( 195 .6 ) .

To d e s c r ib e t h e i r cohom ology, S erre in tro d u c e d c e r t a in

fundam ental homomorphisms c a lle d the B o c k s te in o p e r a t io n s . th e s e , i t

,

i s c o n ven ien t t o take a v e r y Say

e , C1

a re two a b e lia n c a t e g o r ie s , and

F : C -*■

i s a l e f t e x a c t fu n c to r w ith d e r iv e d fu n c to r s

Assume th a t

a) b)

£ z+

To understand

g e n e r a l f u n c t o r i a l s e t t in g :

R1F.

and

s u r je c t iv e homomorphisms form an in v e r s e system . 193

An “ * A n , ,

a ll

n’ < n

1 9k

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

Let AQ = (0 ), and AQ AQ be the 0 homomorphism. Let Kq n t be the kernel of AR -► AQt. Then there is a spectral sequence, with Eh q

(Warning:

this

p

=

RP+q

f ( k p+ i

,p )

is not the characteristic of

•

k.)

In fact, if BP ,q

-

Z?’q

-

Im

R?+q p ( V r , P } ^ rP+x})

.

THE FUNDAMENTAL THEOREM IN CHARACTERISTIC

p

195

In p a r t ic u la r , E °’ q

and

Z ° ,q

=

B ^ X , ffl,1>x)

=

H ^ X , Oy)

i s th e subgroup o f D e fin itio n :

H ^ X , £x )

w hich l i f t s

The homomorphisms

c a lle d the B o c k s te in o p e r a t i cns

dp

to

H ^ X , Bp x ) .

Z^,q C H^CX, Oy)

on

a re

.

The p o in t i s : (* )

n k er (P r )

= f x € H ^ X , ox ) |x l i f t s

r

I

t o H ^ X , Br X)1

fo r a l l

r

’

J

To have a b e t t e r u n d erstan d in g o f t h is a p p aratu s, we need one more f a c t : K er P r o o f:

“* \ ,X ]

( ®n+ i , X

T h is f o llo w s

s

•

im m ed ia tely from th e co rre s p o n d in g r e s u lt

on W itt r in g schemes, v i z . , the k e r n e l o f the tr u n c a tio n : ff( i , p , p 2, . . . , p n }

"•

* { i , p , P a , . . . , P n- 1)

i s isom orph ic, as a d d it iv e group scheme, t o Lectu re 2 6 , §3D (a )

A1 .

T h is was remarked in

(ta k e V n )• P

T h e r e fo r e ,

QED

pr+1

i s a c a n o n ic a l homomorphism:

Ker O r )

H5L+1 (X , Oy) /im O r )

-

n I # (X , ox )

2 °.

Let

F

•

be a n o n -s in g u la r p r o j e c t i v e s u rfa c e o v e r

(a c t u a lly n e ith e r the n o n - s in g u la r it y , nor the dim ension b e in g s e n t ia l) .

2

k is es­

We can now p ro ve th e fundam ental theorem c o n cern in g the f ami H e

o f curves on

F when

c h a r(k )

= p.

Let

P

be th e connected component o f

We know from L e c tu re 2k th a t the

the id e n t it y in the P ic a r d scheme o f

F.

tangent space

c a n o n ic a lly isom orp h ic t o

TQ p

to

v ia th is id e n t ific a t io n r -

P

at

0

is

H1 (F , Op)

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE

196

THEOREM: space o f

The ta n gen t space t o

H1 (F , Op) P r o o f:

t e T^ o ,r

Let

I (n ) and l e t

t

Pped

correspon ds to the sub­

a n n ih ila te d by a l l the B o c k s te in o p e r a tio n s . Let = Spec ktsl/C ® 11)

>

correspon d t o the homomorphism

h2: a)

t

is

I (2 )

ta n gen t t o

lifts

* p

Ppec^ i f

t o a morphism

anh o n ly i f ,

fo r a l l

n,

h2

h^:

h2

I (2)- ± y V n

/

T

^

^

(n ) P roof o f a ): le t

h2

and

t

In terms o f l o c a l r in g s ,

f p: ft C v

Let it

r — ►k [e ] /e2

be th e i d e a l o f n ilp o t n e t s .

fo llo w s th a t

le t

v = Oq p ,

and

d e fin e

f 2( ft ) = 0.

in ( A ) , L e c tu re 22,

f g lifts

Sin ce to

v ----- ►u/ft

Then i f

u/ft

f

. t

i s ta n gen t t o

Ppecp

i s r e g u la r , by the P r o p o s it io n

:

p — -— ►k [ 6 ] / ( e 2)

fn X k [e ] / ( e ) hence

h^.

C o n v e rs e ly , i f

fo r every

h2

n, t o an

f

Suppose

Let

= a • e,

a € k.

f 2(x )

lifts

to

.

x € ft ;

h2 l i f t s then

to x 01 = 0

then

f 2 lifts ,

f o r some

m.

Then

0 =

+

= ^ m + l^ x ^ = [a • e + . . . ] m

T h e re fo re i.e .,

t

c P = 0, is

hence

tan gen t to

a = o. Pped •

T h is means th a t

f2

a n n ih ila t e s

ft,

THE FUNDAMENTAL THEOREM IN CHARACTERISTIC Now t r a n s la t e t h is i n t o fu n c to r s : H a m (I,ns, L (n ), P) r/

C H o m ((n)J I,-, u nuiiHx

fo r a l l

p

197

n,

II P ( | ) )

2II

;n h 1( f

, oi ) ^ y i (n ) 2II

H1(F , But

[Op 0 k [s ]/ e n ] * ^

H1 (F ,

oF • [1 + Op 0 ( e ) / ( s n) ]

e .

i d e a l g e n era ted b y

(o p 0 k [e ]/ 6 n) * )

.

where

(e )

d en otes the

T h e re fo re

[o p ® k [e ]/ e n ] * )

s H1 (F , oj!,) ® H1 (F ,

U op ® - ^ - )

.

I t f o llo w s t h a t : Subgroup o f

I ^ - v a l u e d p o in ts o f

P

at

0

211

H1 (F ,

1 + Op ®

) (e )

211

H1 (F , < V° >p )

Now we use the r e s u lt s o f L e c tu re 2 6 , (E ) .

We

r in g scheme o v e r a f i e l d

p,

is in v e r tib le .

T h e r e fo r e

o f c h a r a c t e r is t ic w

decomposes as

Q, = a l l prim es but

b)

p

ft

p

( E ) ,w ith

P = { 1 , p , p 2, . . . , )

P = £p)

T h e r e fo r e , i f

in

a re w ork in g w ith the W it t so e v e r y prim e e x c e p t

< n-1

and

p; p

0 4.1

Q = in t e g e r s prim e to > n,

we

p .

g e t:

V ia the tr u n c a tio n :

Wd , 2 , . . . , n - 1 5 x S p ec(k )

-

w( 1, p , p 2, ___,p ^ } * sPe 0 ( lc)

the l a t t e r r i n g scheme i s a d i r e c t summand o f th e fo rm er. T h e r e fo r e , f o r e v e r y

n,

we g e t a diagram :

LECTURES ON CURVES ON AN ALGEBRAIC SURFACE I^ - v a lu e d

i

ss

p o in ts o f P j at o

H1 (F , < * CT, 2 , . . . , n - i }> p >

,

Y

H