Algorithmic Aspects of Hyperelliptic Curves

8/13/2019 Algorithmic Aspects of Hyperelliptic Curves

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UNIVERSIT DAIX-MARSEILLEFacult des Sciences de Luminy

THSEpour obtenir le grade de

DOCTEUR DE LUNIVERSIT DAIX-MARSEILLEet, dans le cadre dune Cotutelle, de PhD de lUniversit de Sydney

cole Doctorale en Mathmatiques et Informatique de MarseilleDiscipline: Mathmatiques

prsente et soutenue publiquement par

Hamish IVEY-LAW

le 14 dcembre 2012

Titre:

Algorithmic Aspects ofHyperelliptic Curves and their Jacobians

Les directeurs de thse: David KOHEL et Claus FIEKERLes rapporteurs: Kamal KHURI-MAKDISI et Michael STOLL

JURY:Dr. Claus FIEKER Directeur de thseDr. Florian HESS ExaminateurDr. David KOHEL Directeur de thseDr. David LUBICZ ExaminateurDr. Christophe RITZENTHALER ExaminateurDr. Michael STOLL Rapporteur


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Abstract (English)

The contribution of this thesis is divided naturally into two parts. In PartIwe generalise the work of Khuri-Makdisi [22] on algorithms for divisor arith-

metic on curves over fields to more general bases. We prove that the naturalanalogues of the results of Khuri-Makdisi [22] continue to hold for relativeeffective Cartier divisors on projective schemes which are smooth of rela-tive dimension one over an arbitrary affine Noetherian base scheme and thatnatural analogues of the algorithms remain valid in this context for a cer-tain class of base rings. We introduce a formalism for such rings, which arecharacterised by the existence of a certain subset of the usual linear algebraoperations for projective modules over these rings.

PartIIof this thesis is concerned with a type of Riemann-Roch problemfor divisors on certain algebraic surfaces. Specifically we consider algebraic

surfaces arising as the square or the symmetric square of a hyperelliptic curveof genus at least two over an (almost) arbitrary field. The main results area decomposition of the spaces of global sections of certain divisors on suchsurfaces and explicit formul for the dimensions of the spaces of sections ofthese divisors. In the final chapter we present an algorithm which generatesa basis for the space of global sections of such a divisor.

We now explain the content of each part of the thesis in more detail.

Relative curves and their Jacobians

Khuri-Makdisi [22]introduced a means to perform arithmetic with divisorson curves over fields which represents a divisor as a subspace of the space ofglobal sections of some power of given very ample divisor. On the basis oftwo principal theoretical results, he shows that many natural operations ondivisors can be reduced to relatively simple linear algebra over the base field.Chapters2 and 3 of this thesis are concerned with the generalisation of theresults of Khuri-Makdisi [22].

Let S = Spec(R) be an affine Noetherian scheme and let X/S be aprojective S-scheme which is smooth of relative dimension one over S; we


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callX/Sarelative curve. In Section2.1we show that the modules of sections

of a relative effective Cartier divisor on Xare projective. In Section2.2 wedescribe conditions for very ampleness and normal generation of invertiblesheaves. In particular we show that the usual relationships between veryampleness and normal generation can be lifted from the fibres, as can thefamiliar result that a divisor of degree at least 2g+ 1is normally generatedand hence very ample. In Section2.3we prove the two principal results thatunderlie the algorithms for performing arithmetic with divisors:

Proposition 2.3.5. LetS = Spec(R), letX/Sbe a relative curve and letM andN be normally generated sheaves onX. Then

: H0(X,M) H0(X,N) H0(X,MN)

is surjective.

Proposition2.3.7. LetX/Sbe a relative curve of genusg and letM andN be invertible sheaves onX, each of degree at least2g+ 1. Then for anyrelative effective Cartier divisorD onXof degree at mostdeg(M)(2g +1),we have

H0(X,M(D)) =

H0(X,MOX N(D)) :H0(X,N)

with respect to the canonical homomorphismof Proposition2.3.5.

In Chapter3we demonstrate that Propositions2.3.5and2.3.7allow usto derive algorithms for arithmetic of divisors which are analogous to thosepresented by Khuri-Makdisi [22]. We reprove a selection of his algorithmsin this more general context; in particular, we give algorithms that allowaddition of divisor classes in Pic0X(S).

Spaces of sections on algebraic surfaces

Let C be a hyperelliptic curve of genus g 2, and let Sym2(C) be the

symmetric product of C. Let C(k) be a Weierstrass point over analgebraic closure of the base field k and set D= 2(). Letpi : C C Cfor i = 1, 2be the two projection maps. Let F = p1(D) + p

2(D)be the

fibral divisor on C Cand let be the antidiagonal divisor on C C. Set=g 1and let m > and r 0. The main results of Chapter4 are thefollowing. In Theorem4.5.11we prove that, under mild conditions, there isa decomposition

H0(C C,mF+ r) =H0(C2, mF) r

i=1

H0(C, (2m i)D).


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Rsum (franais)

Ce travail se divise en deux parties. Dans la premire partie, nous gnral-isons le travail de Khuri-Makdisi [22]qui dcrit des algorithmes pour larith-

mtique des diviseurs sur une courbe sur un corps. Nous montrons que lesanalogues naturelles de ses rsultats se vrifient pour les diviseurs de Cartierrelatifs effectifs sur un schma projectif, lisse et de dimension relative unsur un schma affine noetherien quelconque, et que les analogues naturellesde ses algorithmes se vrifient pour une certaine classe danneaux de base.Nous prsentons un formalisme pour tels anneaux qui sont caractriss parlexistence dun certain sous-ensemble des oprations standards de lalgbrelinaire pour les modules projectifs sur ces anneaux.

Dans la deuxime partie de ce travail, nous considrons un type de prob-lme de Riemann-Roch pour les diviseurs sur certaines surfaces algbriques.

Plus prcisment, nous analysons les surfaces algbriques qui manent dunproduit ou dun produit symtrique dune courbe hyperelliptique de genresuprieur un sur un corps (presque) arbitraire. Les rsultats principauxsont une dcomposition des espaces de sections globales de certains diviseurssur telles surfaces et des formules explicites qui dcrivent les dimensions desespaces de sections de ces diviseurs. Dans le dernier chapitre, nous prsen-tons un algorithme qui produit une base pour lespace de sections globalesdun tel diviseur.

prsent, nous dcrivons de manire plus dtaille le contenu de chaquepartie de la thse.

Courbes relatives et leurs jacobiennes

Khuri-Makdisi [22] a prsent un moyen de calculer certaines oprationsarithmtiques avec les diviseurs sur une courbe sur un corps fini qui reprsenteun diviseur comme un sous-espace de sections globales dune puissance dundiviseur donn trs ample. partir de deux rsultats thoriques principaux,il a montr que plusieurs oprations naturelles sur les diviseurs peuvent trerduites lalgbre linaire sur le corps de base. Les Chapitres2et3sont


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consacrs la gnralisation des rsultats de Khuri-Makdisi[22].

Soient S = Spec(R) un schema affine noetherien et X/S un S-schemaprojectif, lisse de dimension relatif un sur S; on appelle X/S une courberelative. Dans la Section2.1, nous montrons que les modules de sectionsdun diviseur de Cartier relatif effectif sur X sont projectifs. Dans la Sec-tion2.2,nous dcrivons les conditions pour quun faisceau inversible soit trsample ou engendr normalement. En particulier, nous montrons que les re-lations attendues entre les proprits dtre trs ample ou dtre engendrnormalement se relvent des fibres. Dans la Section2.3, nous montrons lesdeux rsultats principaux qui sous-tendent les algorithmes pour calculer lesoprations arithmtiques sur les diviseurs :

Proposition 2.3.5. Soient S = Spec(R) et X/S une courbe relative, etsoientM etN deuxOX-modules qui sont engendrs normalement. Alors

: H0(X,M) H0(X,N) H0(X,MN)

est surjective.

Proposition2.3.7. SoitX/Sune courbe relative de genreget soientM etNdeux faisceaux inversibles surX, chacun de degr au moins2g + 1. Alorstout diviseur de Cartier relatif effectifDsurXde degr au plusdeg(M)

(2g+ 1) satisfait lquationH0(X,M(D)) =

H0(X,MOX N(D)) :H

0(X,N)

par rapport lhomomorphisme canoniquede Proposition2.3.5.

Dans le Chapitre3nous montrons que les Propositions2.3.5et2.3.7nouspermettent de dcrire des algorithmes pour larithmtique des diviseurs quisont analogues ceux prsents par Khuri-Makdisi [22]. Nous drivons nouveau une slection de ses algorithmes dans ce contexte plus gnral ; enparticulier nous donnons un algorithme qui calcule la somme de deux classes

de diviseurs dans Pic0X(S).

Espaces de sections sur une surface algbrique

Soient Cune courbe hyperelliptique de genre g 2 et Sym2(C) le produitsymtrique de C. Soient C(k) un point Weierstrass sur une clturealgbrique du corps de base k et D = 2(). Soient pi: C C Cpouri= 1, 2les deux applications de projection. Soient F =p1(D) +p

2(D)le

diviseur fibral sur C C et le diviseur anti-diagonal sur C C. Notons


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=g 1et soient m > et r 0. Les rsultats principaux du Chapitre 4

sont les suivants. Dans le Thorme4.5.11, nous montrons, sous certainesconditions lgres, quil existe une dcomposition

H0(C C,mF+ r) =H0(C2, mF) r

i=1

H0(C, (2m i)D).

On obtient comme corollaire des formules qui dcrivent la dimension de cetespace comme une fonction de met r. Par exemple, lorsque 2m r > ,on obtient

dim H0(C C,mF+ r) = (2m )2 + 4mr r(r+ 2).

Dune faon similaire, dans le Thorme4.6.12nous montrons, sous certainesconditions lgres, quil existe une dcomposition

H0(Sym2(C), 2mS+rS) =H0(Sym2(C), 2mS)

ri=1

H0(P1, (2mi)())

o 2S et S sont les images de F et sous la projection canoniqueC C Sym2(C). Comme corollaire, on obtient des formules qui dcriventla dimension de cet espace en fonction de m et r. Par exemple, lorsque

2m r >0, on obtient

dim H0(Sym2(C), 2mS+ rS) =

2m + 1

2

+ r(2m + 1)

r+ 1

2

.

Nous ne pouvons montrer les rsultats dans le cas o 2m r = 0. Nan-moins, nous conjecturons que les rsultats se vrifient aussi dans ce cas. Eneffet, lalgorithme prsent dans le Chapitre 5 nous permet de vrifier laconjecture dans chaque cas particulier.

Dans le Chapitre5,nous dcrivons une dcomposition deH0(C C,mF)par rapport aux espaces propres de laction de la permutation de coordonnes

et nous nous en servons pour dvelopper un algorithme qui produit unebase explicite pour lespace H0(Sym2(C), 2mS+ rS). Nous montronsque lalgorithme reproduit, modulo une transformation projective linaire,la base bien connue de Cassels [7]et Flynn[12] qui dcrit le plongement dela jacobienne dune courbe de genre deux dans P15. Nous montrons de plusquil vrifie les formules de dimension du Chapitre 4 dans plusieurs cas. Onconclut avec une discussion brve des applications potentielles la thoriedes codes sur les surfaces de la forme C Cet Sym2(C).


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Dedicated to the memory ofPhyllis M. Ivey

19232010


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Contents

1 Geometric preliminaries 1

1.1 Divisors and invertible sheaves. . . . . . . . . . . . . . . . . . 11.2 Cohomology of sheaves . . . . . . . . . . . . . . . . . . . . . . 31.3 Embeddings in projective space . . . . . . . . . . . . . . . . . 51.4 The Picard group . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Quotient varieties . . . . . . . . . . . . . . . . . . . . . . . . . 101.6 Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Hyperelliptic curves. . . . . . . . . . . . . . . . . . . . . . . . 13

I Relative curves and their Jacobians 16

2 Divisors on relative curves 172.1 Relative effective Cartier divisors . . . . . . . . . . . . . . . . 172.2 Criteria for very ampleness. . . . . . . . . . . . . . . . . . . . 212.3 Tensor products and sheaf quotients. . . . . . . . . . . . . . . 24

3 Divisor arithmetic on relative Jacobians 293.1 Complexity analysis. . . . . . . . . . . . . . . . . . . . . . . . 293.2 Amenable rings . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Arithmetic of modules . . . . . . . . . . . . . . . . . . . . . . 323.4 Tensor products and sheaf quotients. . . . . . . . . . . . . . . 35

3.5 Arithmetic of divisors. . . . . . . . . . . . . . . . . . . . . . . 373.6 Arithmetic on a relative Jacobian . . . . . . . . . . . . . . . . 39

II Spaces of sections on algebraic surfaces 43

4 Cohomology of surfaces 444.1 Intersection theory on surfaces . . . . . . . . . . . . . . . . . . 444.2 Equivalence classes of divisors . . . . . . . . . . . . . . . . . . 46


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CONTENTS x

4.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Cohomology of divisors onJC . . . . . . . . . . . . . . . . . . 534.5 Cohomology of divisors onC C . . . . . . . . . . . . . . . . 554.6 Cohomology of divisors onSym2(C). . . . . . . . . . . . . . . 62

5 Explicit bases of sections and applications 715.1 Eigenspace decompositions . . . . . . . . . . . . . . . . . . . . 715.2 Hasse-Schmidt derivations . . . . . . . . . . . . . . . . . . . . 735.3 Formal neighbourhoods of . . . . . . . . . . . . . . . . . . . 745.4 Explicit bases of sections . . . . . . . . . . . . . . . . . . . . . 775.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5.1 Projective embeddings . . . . . . . . . . . . . . . . . . 795.5.2 Conjecture4.5.10 . . . . . . . . . . . . . . . . . . . . . 795.5.3 Codes on surfaces . . . . . . . . . . . . . . . . . . . . . 80

Bibliography 81


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Chapter 1

Geometric preliminaries

In this chapter we recall the background material and notation necessary forthe subsequent chapters. Most of the material presented in this chapter canbe found in Hartshorne [17], Liu[26] or Mumford [32].

1.1 Divisors and invertible sheaves

Let (X,OX) be a scheme and let KXbe the sheaf of total quotient rings(also called the sheaf of stalks of meromorphic functions) on X. We denote

by KXthe sheaf of multiplicative groups of invertible elements of the sheafof rings KX. Similarly, we denote by OX the sheaf of groups of invertibleelements of the sheaf of rings OX. For any sheafFonX, we will denote theglobal sections ofFby(X,F).

ACartier divisoronXis a global section of the sheafKX/OX. A Cartier

divisor can therefore be given as a system {(Ui, fi)}iI where {Ui}iI is anopen covering ofXand the functionsfiare sections of(Ui,KX)such thatfor all (i, j) I I, we have fi/fj (Ui Uj ,OX); that is, the ratio fi/fjis a non-zero regular function on Ui Uj. Two such systems {(Ui, fi)}iIand{(Vj , gj)}jJ define the same Cartier divisor if fi/gj OX(Ui Vj) for all

(i, j) I J.The set of Cartier divisors(X,KX/OX)on Xwill be denoted byDiv(X);

it has a group structure defined as follows. Let D1 = {(Ui, fi)}iI andD2 ={(Vj, gj)}jJbe two Cartier divisors on X. Then the sum D1+D2 isgiven by {(Ui Vj, figj)}(i,j)IJ, the negative D1is given by {(Ui, f1i )}iI,and the identity element is {(X, 1)}.

The image of a non-zero rational function f (X,KX)in(X,K

X/OX)

is called a principal Cartier divisorand will be denoted by div(f). TwoCartier divisorsD1and D2are said to berationally equivalentif their differ-


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1.2 Cohomology of sheaves 3

1.2 Cohomology of sheaves

Let Xbe a scheme and let Fbe a sheaf onX. We denote byHi(X,F)theith cohomology group ofF. The definition ofHi(X,F)is beyond the scopeof this work; see Hartshorne[17, Chapter III] or Liu [26, Chapter 5] for thedefinition. We will collect here the main properties of the cohomology groupsof coherent sheaves on projective varieties.

Proposition 1.2.1.LetXbe a scheme and letFbe a sheaf onX. Then

H0(X,F) =(X,F).

Proof. See Hartshorne [17,Section III.2] or Liu [26,Proposition 5.2.6].

Proposition 1.2.2.LetXbe a scheme and let

0 F F F 0

be a short exact sequence of sheaves on X. Then for all i 0 there existcoboundary maps

i : Hi(X,F) Hi+1(X,F)

giving a long exact sequence of cohomology groups

0 H0(X,F) H0(X,F) H0(X,F)

0

H1(X,F) H1(X,F) H1(X,F) 1

(1.1)

Proof. See Hartshorne [17,Section III.2].

Theorem 1.2.3. Let X be a projective scheme over a Noetherian ring Aand letFbe a coherent sheaf onX. ThenHi(X,F)is a finitely generatedA-module for alli 0.

Proof. See Hartshorne [17,Theorem III.5.2(a)].

Theorem 1.2.4. LetXbe a projective variety of dimensionnover an alge-braically closed fieldk and letFbe a coherent sheaf onX. Then

(i) dimkHi(X,F) = 0fori > n.

(ii) (Serre Duality) Suppose moreover that Xis nonsingular and thatFis locally free; writeF = HomOX(F,OX). Then for all i, there is anatural isomorphism

Hi(X,F) =Hni(X,F OX X)

whereX = nX =n 1Xdenotes the canonical sheaf on X, and V

denotes the dual of a finite vector spaceV.


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1.2 Cohomology of sheaves 4

Proof. See Hartshorne [17, Theorem III.2.7] for (i) and Hartshorne[17, Corol-

lary III.7.7 and Corollary III.7.12] for (ii).

We will often write hi(X,F) = dimkHi(X,F) when the base field isclear from the context.

Theorem 1.2.5 (Knneth Formula). Let X and Y be projective varietiesand letF andGbe quasi-coherent sheaves onX andY respectively. Then

for alli 0,

Hi(X Y, p1Fp2G)

=

j+k=i

Hj(X,F) Hk(Y, G)

wherep1 : X Y Xandp2 : X Y Yare the projection maps.

Proof. See Kempf[21,Proposition 9.2.4].

Recall that a morphism f :X Yof schemes is called affine if thereexists a covering of Y by open affine sets {Vi} such that f1(Vi) X isaffine for all i; this is equivalent to the condition that, for anyopen affinesubset V Y, the open set f1(V) is affine (see Hartshorne [17, ExerciseII.5.17(a)]).

Proposition 1.2.6.Letf: X Ybe a morphism of schemes and letF be

a sheaf onX. For alli 0, there is a canonical homomorphism

Hi(Y, fF) Hi(X,F)

which is an isomorphism ifFis quasi-coherent and either (i)Xis separated,andfis affine, or (ii) fis a closed immersion.

Proof. See Liu [26,Exercise 5.2.3].

LetXbe a projective variety over a field k and let Fbe a coherent sheafon X. TheEuler-Poincar characteristicofFis defined to be

(F) =i0

(1)ihi(X,F).

By Theorems 1.2.3 and 1.2.4(i the Euler-Poincar characteristic is alwaysfinite. In particular, ifXis a curve, then Theorem1.2.4(i)

(F) =h0(X,F) h1(X,F).

IfL = OX(D)for some divisor D on X, then we will write (D)for (L).


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1.3 Embeddings in projective space 5

Theorem 1.2.7(Riemann-Roch). LetXbe a projective curve over a fieldk

and letLbe an invertible sheaf onX. Then there exists an integerdeg(L)such that

(L) = deg(L) + (OX) = deg(L) + 1 g.

Proof. See Liu[26, Theorem 7.3.17 and Theorem 7.3.26] or Hartshorne [17,Theorem IV.1.3].

Let Xbe a projective curve over a field k. The degreeof an invertiblesheafL on Xis defined to be the integer

deg(L) =(L) (OX).

Note that, ifL = OX(D)for a divisor Don X, then deg(L) = deg(D).

Corollary 1.2.8. LetXbe a projective curve over a fieldk and letLbe aninvertible sheaf onX. Ifdeg(L)> 2g 2, then

dim(X,L) = deg(L) + 1 g.

Proof. See Liu [26,Remark 7.3.33].

1.3 Embeddings in projective space

Let S =

dZ Sd be a graded ring and let X = Proj(S). For any integern, letS(n)denote the twisted module

dZ Sd+n. The OX-module OX(n)is

defined to be the sheaf associated to the graded module S(n). When n = 1,we call OX(1) the twisting sheaf of Serre. For an arbitrary OX-module F,we define thenth twist ofFto be F(n) = FOX OX(n). For simplicity, forany sheafF, we will write Fn for Fn = F F(ntimes) throughoutthis work.

Proposition 1.3.1. LetS=

dZ Sdbe a graded ring which is generated by

S1as anS

0-algebra and letX= Proj(S). Then we have the following:

(i) The sheafOX(n)is invertible onXfor alln.

(ii) For allmandn,

OX(m) OX OX(n)= OX(m + n).

In particular, for any sheaf ofOX-modulesF

F(m) OX OX(n)= F(m + n).


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(iii) LetT =dZ Td be another graded ring which is generated byT1 as aT0-algebra and letY= Proj(T). Let : S Tbe a homomorphism ofgraded rings (so degrees are preserved), letU Ybe the maximal opensubset ofYsuch thatf: U Xis a morphism determined by. Then

f(OX(n)) = OY(n)|U and f(OY(n)|U) =(fOU)(n).

Proof. See Hartshorne [17,Proposition II.5.12].

The gradedS-module associated to an OX-module Fis defined to be thegroup

(F) = nZ (X,F(n)).This group is given the structure of a graded S-module as follows. Anyelements Sddetermines a global sections (X,OX(d)). Then given anyglobal section t (X,F(n)), the product s t (X,F(n+d)) is givenby the image ofs tunder the isomorphism F(n) OX OX(d) = F(n+d)of Proposition1.3.1(ii). In particular, for any ringA, we have (OPr

A) =

A[x0, . . . , xr](see Hartshorne [17,Proposition II.5.13]).Let Lbe an invertible sheaf on a Y-scheme X Y. Then L is said

to bevery amplewith respect to Y if there is an immersion : X PrY forsome r 1such that (OPr

Y(1)) = L. IfL and Mare very ample sheaves

on Xwith respect to Y, then LOX Mis also very ample (see Hartshorne[17, Exercise II.5.12]).

Let Fbe an OX-module on a schemeX. Then Fis said to begeneratedby global sectionsif there is a set of global sections

{s(i)}iI (X,F)

such that, for all x X, the stalk Fx ofF at x is generated as an OX,x-module by {s(i)x }, where s

(i)x denotes the image ofs

(i) in Fx. For example,ifS=

dZ Sd is a graded ring which is generated by S1 as an S0-algebra,

and if X = Proj(S), then OX(1) is generated by S1 (X,OX(1)) (see

Hartshorne [17, Example II.5.16.3]).Let Xbe a projective scheme over a Noetherian ring A and let L be

an invertible sheaf on X. Then Lis said to be ampleif, for every coherentOX-module F, there is an integer n0 such that, for all n n0, the sheafFOX L

n can be generated by a finite number of global sections.

Theorem 1.3.2(Serre).LetLbe a very ample sheaf on a projective schemeXover a Noetherian ringA. ThenL is ample.

Proof. See Hartshorne [17,Theorem II.5.17].


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Theorem 1.3.3. LetXbe a scheme of finite type over a Noetherian ringA

and letLbe an invertible sheaf onX. ThenLis ample if and only ifLmis very ample overSpec(A)for somem >0.


Theorem 1.3.4. Let A be a ring, let X be an A-scheme, and let PrA =Proj(A[x0, . . . , xr])be projectiver-space overA.

(i) If : X PrA is a morphism of A-schemes, then (OPr

A(1)) is an

invertible sheaf on X which is generated by the global sections si =(xi)fori= 0, . . . , r.

(ii) Conversely, supposes0, . . . , sr (X,L)are global sections which gen-erate an invertible sheafL onX. Then there exists a unique morphism : X PrA ofA-schemes such thatL =

(OPrA

(1))andsi=(xi).


Corollary 1.3.5. A very ample sheaf is generated by global sections.

Proof. This follows directly from Theorem1.3.4.

Proposition 1.3.6. LetXbe a smooth projective curve of genusg over a

fieldk. LetLbe an invertible sheaf onX.

(i) Ifdeg(L) 2g, thenLis generated by global sections.

(ii) Ifdeg(L) 2g+ 1, thenL is very ample.

Proof. See Hartshorne [17,Corollary IV.3.2].

Proposition 1.3.7.LetAbe a ring.

(i) If Y is a closed subscheme of PrA, then there is a homogeneous idealI A[x0, . . . , xr]such thatY is the closed subscheme determined byI.

(ii) A schemeY overSpec(A) is projective if and only if it is isomorphicto Proj(S) for some graded ringS=

dZ Sd, whereS0 =AandS is

finitely generated byS1 as anS0-algebra.

Proof. See Hartshorne [17,Corollary II.5.16].


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1.4 The Picard group 8

LetAbe a ring and letXbe a closed subscheme ofPrA. Thehomogeneous

coordinate ringS(X)ofXfor the given embedding is defined to be

S(X) =A[x0, . . . , xr]/I

where Iis the homogeneous ideal (IX)associated to the ideal sheafIXofX. A scheme X isnormalif its local rings are integrally closed domains.A closed subscheme X PrA isprojectively normalfor the given embeddingif its homogeneous coordinate ring S(X)is an integrally closed domain.

Let Lbe an ample invertible sheaf on a scheme X. Then L is said tobe normally generated if

(X,L)n

(X,Ln

)

is surjective for alln 1. This is equivalent to the condition that

(X,Li) k(X,Lj) (X,L(i+j))

is surjective for alli, j 1.

Theorem 1.3.8. LetXbe smooth, complete curve of genusg over an alge-braically closed fieldk. LetM andN be invertible sheaves onXsuch thatdeg(M) 2g+ 1 anddeg(N) 2g. Then the map

(X,M

) k(X,N

) (X,M

OXN

)is surjective.

Proof. See Mumford[31, Theorem 2.6].

Corollary 1.3.9. Let X be a smooth, complete curve of genus g over analgebraically closed field k. If L is an invertible sheaf on X of degreedeg(L) 2g+ 1, thenLis normally generated.

Proof. See Mumford[31, Corollary to Theorem 2.6].

1.4 The Picard groupProposition 1.4.1. IfL andMare invertible sheaves on a ringed spaceX, then so isLOX M. LetLbe an invertible sheaf onXand define

L1 = HomOX(L,OX)

(this is the dual sheaf ofL). ThenL1 is an invertible sheaf onXand

L OX L1 = OX.


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1.5 Quotient varieties 10

1.5 Quotient varieties

Theorem 1.5.1. LetXbe an variety over an algebraically closed fieldk andletGbe a finite subgroup ofAut(X). Suppose that, for anyx X, the orbitofx is contained in an affine open subset ofX. Then there is a varietyYand a morphism : X Ysatisfying the following properties:

(i) As a topological space, Y is the quotient ofXfor the action ofG; thatis, Y =X/Gas a set andU Yis open if and only if1(U)is open.

(ii) For any open affine setU Y, we have

(U,OY) = (1(U),OX)

G

where(1(U),OX)G denotes the elements of(1(U),OX)fixed bythe action ofG.

These conditions determine the pair (Y, ) up to isomorphism. The mor-phism is finite, surjective and separable. IfXis affine, then so isY.

Proof. See Mumford[32, Theorem 7.1].

The pair (Y, ) appearing in Theorem1.5.1is called the geometric quo-tientofXbyG.

Let G Aut(X)be a finite group acting on X, let : X X/Gbe the

quotient, and letF

be a coherent sheaf on X/G. For any g G, we have = g and it follows that g induces an automorphism g : F F.Hence G acts on Fin a manner compatible with the action on X. Wedefine a coherentG-sheaf onXto be a coherent OX-module onXon whichGacts in a manner compatible with the action on X.

Let Fbe a sheaf on X. Recall that the direct image ofFis defined by(V, F) = (

1(V),F)for all open V X/G. IfF is a G-sheaf on X,then, for all open V X/G, the group Gacts on (1(V),F)and henceF is a G-sheaf on X/Gsince Gacts trivially on X/G. For any G-sheafF on Xwe denote by (F)G the sheaf on X/Gdefined by

V (V, F)G

= (1

(V),F)G

.The following result is a modification of Mumford [32, Proposition 7.2]

where we have removed the condition that Gact freely, but we impose thecondition that Lbe locally free.

Proposition 1.5.2.Let : X X/Gbe the geometric quotient of a varietyXby a finite group of automorphismsG, and letLbe a locally free sheaf onX/G. ThenL =(L)G and, in particular,

(X/G,L) =(X, L)G.


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1.6 Abelian varieties 11

Proof. We first reduce to the affine case as in the proof of Mumford [32, The-

orem 7.1]. For anyxinX, letU be an open affine subset ofXcontaining theorbit Gxofx. Then U=

gG gU

is an affine open subset ofXcontainingxwhich is stable under the action ofG. Thus Xhas a covering ofG-stableaffine open sets U. If the proposition holds for X and X/G affine, thenL(U) =(L)G(U)for this covering and consequently L =(L)G.

Now assume X = Spec(A) is affine, so X/G = Spec(B) with B = AG,and let (X/G,L) =M. We have

(X/G, (L)G) = (X, L)G=(MBA)

G=MG,

so it suffices to prove that M = MG

. As M is locally free, we can assumethat B is local and Mis free, in which case the result follows from the factthat (Br)G = (BG)r =Br for anyr 1.

Let Xbe a variety. The symmetric group Sn in n letters acts on then coordinates of the n-fold product Xn = X Xby permutation.We define the nth symmetric productofX, denoted by Symn(X), to be thegeometric quotient ofXn bySn; the symmetric product exists by Theorem1.5.1. IfXis a smooth curve, then Symn(X)will be smooth. Moreover theeffective divisors of degreedcan be identified with points on Symd(X).

1.6 Abelian varietiesAn abelian variety is a complete group variety. An abelian variety of di-mension one is called an elliptic curveand an abelian variety of dimensiontwo is called an abelian surface. The group law on an abelian variety is al-ways commutative (see Milne [27,Corollary 2.4]) and every abelian varietyis projective (see Milne [27, Theorem 7.1]). Amorphismof abelian varieties: A B is a morphism of varieties that is also a group homomorphismwith respect to the group laws ofAand B. Ifis finite and surjective it iscalled anisogeny, and two abelian varieties are said to be isogenousif there

exists an isogeny between them.Let Lbe an invertible sheaf on an abelian variety A. To Lwe associate

a map L :A Pic(A) defined by x xLL1, where x: A Ais

the translation-by-xmap on A. We define KLto be

KL = {x A | xL

= L1};

it is a reduced closed subscheme ofA(see Milne [27, Section 9]). IfL =OA(D)for a divisor Don A, we write D for L and KD for KL.


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1.6 Abelian varieties 12

Theorem 1.6.1. LetAbe an abelian variety of dimensiong, and letL be

an invertible sheaf onA. Then

(i) The degree ofL is(L)2.

(ii) Ifdim KL = 0, then there exists a unique integerr = r(L),0 r g,such thatHi(A,L) = 0fori =r andHr(A,L) = 0.

Proof. See Milne [27,Theorem 13.3].

There is a canonical way to attach an abelian variety to a curve, asdescribed in the following theorem.

Theorem 1.6.2.LetXbe a smooth projective curve of genusg 1over afieldk. Then there exists an abelian varietyJXoverk and an injection

j : X JX

(not necessarily defined overk) with the following properties:

(i) If j is extended linearly to divisors onX, then the induced group ho-momorphism

Pic0(X) JX

is an isomorphism.(ii) LetW0 be the subvariety{0}ofJXand for eachr 1, defineWr to be

the image inJXof the map

Symr(X) JX

defined by

(P1, . . . , P r) r

i=1

j(Pi).

Thendim(Wr) = min{r, g}andWg is birationally equivalent to JX. In

particular,dim(JX) =g.

Proof. See Hindry and Silverman[18, Theorem A.8.1.1] for a sketch of theabove result. A more intrinsic account can be found in the survey of Milne[28], who describes the variety JXas representing the functor Pic

0.

The varietyJXappearing in Theorem1.6.2is called theJacobianof thecurveXand the map j is called theAbel-Jacobi map. Thetheta divisoronJX is the divisor Wg1; it is effective and moreover ample (see Milne [28,Section 6]).


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1.7 Hyperelliptic curves 14

Since deg((g+ 1)D0)> 2g 2, Corollary1.2.8implies that

dim

X,OX((g+ 1)D0)

= deg

(g+ 1)D0

+ 1 g=g + 3.

Thus there exists a function y in (X,OX((g + 1)D0))which is linearly inde-pendent of the g + 2elements {1, x , x2, . . . , xg+1}, and the result follows.

The following proposition shows that the defining equations of a hyper-elliptic curve have a rather rigid form.

Proposition 1.7.2.Letk be a field and letX/k be a hyperelliptic curve ofgenusg.

(i) The function field of X is given by k(X) = k(x)[y] with the definingrelation

y2 + h(x)y= f(x) (1.3)

wherehandfare polynomials ink[x]whose degrees are at mostg+ 1and2g + 2respectively. If, moreover, char(k) = 2, then we can assumethath= 0.

(ii) The curveXis covered by the open affine schemes

U= Spec k[x, y]/y2 + h(x)y f(x)V= Spec k[u, v]/

v2 + h(u)v f(u)

whereh(u) =h(1/u)ug+1 andf(u) =f(1/u)u2g+2 and the two patchesglue along the open sets D(u) D(x) with relations u = 1/x andv = y/xg+1 (where D(u) is the basic open set Spec(OU(U)[1/u]) andsimilarly forD(x)).

Proof. See Liu [26,Proposition 7.4.24].

An equation of the form (1.3) is called anaffine modelfor a hyperelliptic

curve. If

y2 + h(x)y= f(x) and v2 + h(u)v= f(u)

are two affine models for a hyperelliptic curve Xof genus g 2, then thereexists a matrix

a bc d

GL2(k), an element e k, and a polynomial H

k[x]of degree at most g+ 1 such that

u=ax + b

cx + d and v=

H(x) + ey

(cx + d)g+1 (1.4)


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1.7 Hyperelliptic curves 15

(see Liu [26,Corollary 7.4.33(a)]).

LetXbe a hyperelliptic curve over k. The generatorofGal(k(X)/k(x))induces an automorphism of order 2 of X, also denoted by , called thehyperelliptic involutionofX. The hyperelliptic involution ofX is unique ifthe genus ofXis at least2(Liu [26, Proposition 7.4.29]). The non-uniquenessof the rational subfield of degree 2 and the corresponding involution in thecase of elliptic curves is the reason why elliptic curves are not generallysubsumed in the definition of hyperelliptic curves.


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Part I

Relative curves and theirJacobians


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Chapter 2

Divisors on relative curves

In this chapter we prove that natural generalisations of Khuri-Makdisi [22,Lemmas 2.2 and 2.3] continue to hold in the case of relative effective Cartierdivisors on projective schemes which are smooth of relative dimension oneover an arbitrary affine Noetherian base scheme.

Throughout this chapter we fix a scheme S = Spec(R) where R is aNoetherian ring. We use the following notation: let Xbe an S-scheme, letsbe a closed point ofSand let Lbe sheaf on X. Denote the the maximalideal of the local ring OS,s by ms and its residue field by k(s) = OS,s/ms.Denote the fibre ofXabove sby Xs =X SSpec k(s), the projection mapfrom Xsto Xbys : Xs X, and set Ls= sL.

2.1 Relative effective Cartier divisors

We begin by recalling some facts about relative effective Cartier divisors.This material can be found in Katz and Mazur[20, Chapter 1]. Letf: X Sbe an S-scheme. Recall (see Section1.1) that an effective Cartier divisor isa closed subscheme : D X ofXwhose ideal sheaf is invertible. We callDarelative effective Cartier divisor iff : D Sis flat.

The set of relative effective Cartier divisors on X is closed under theusual sum of Cartier divisors. IfT S is another S-scheme, then for anyrelative effective Cartier divisor D on X, the fibre DT =D STis a relativeeffective Cartier divisor on XT =XST. Iff: Y Xis a flat morphism ofS-schemes, thenf(D)is a relative effective Cartier divisor on Y. The usualcorrespondence between isomorphism classes of invertible sheaves and effec-tive Cartier divisors carries over to a correspondence between isomorphismclasses of invertible sheaves flat over Sand relative effective Cartier divisors.


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2.1 Relative effective Cartier divisors 18

Proposition 2.1.1.LetXbe a flatS-scheme of finite type and letDbe an

effective divisor onX. ThenDis a relative effective Cartier divisor onX ifand only if, for all geometric pointsSpec(k) S, the closed subschemeDkis a relative effective Cartier divisor onXk.

Proof. See Katz and Mazur[20, Corollary 1.1.5.2].

We now look more closely at the case where the fibres ofX/Sform a flatfamily of smooth curves. We will call X/Sarelative curveifXis a smoothprojective S-scheme of relative dimension one whose geometric fibres areconnected. ThereforeXcorresponds to a family of geometrically connected,smooth, projective algebraic curves parametrised by S. A relative effective

Cartier divisor on a relative curveX/S is automatically proper over S.

Proposition 2.1.2.LetX/Sbe a relative curve and letLbe an invertiblesheaf onX. Then the mapping

(L) : S Z defined by s (Ls)

is locally constant onS.

Proof. See Grothendieck[15, Thorme 7.4.3].

Proposition2.1.2shows that the genus map gX: S Zsending s gXsof a relative curve is locally constant where gXs denotes the genus of thealgebraic curve Xs. Hence by Riemann-Roch, the degree function deg(L) :S Zdefined by s deg(Ls)is locally constant on S. IfD is a relativeeffective Cartier divisor on X, then deg(D)is given by deg(OX(D)).

Proposition 2.1.3.LetX/Sbe a relative curve and letD1andD2be relativeeffective Cartier divisors onX. Then

deg(D1+ D2) = deg(D1) + deg(D2).

Proof. See Katz and Mazur[20, Lemma 1.2.6].Proposition 2.1.4. Let X/S be a relative curve and let D be a relativeeffective Cartier divisor onX. Then for anyS-schemeT, we have

deg(DT) = deg(D).

Proof. See Katz and Mazur[20, Lemma 1.2.9].


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Proposition 2.1.5. LetX/Sbe a relative curve and letD andD relative

effective Cartier divisors onXsatisfyingD D. Then there exists a relativeeffective Cartier divisor D such that D = D +D and the degree of D

satisfiesdeg(D) = deg(D) deg(D).

Proof. See Katz and Mazur[20, Section 1.3].

Proposition 2.1.6.LetX/Sbe a relative curve and letD andD be relativeeffective Cartier divisors onX satisfyingD D. Then the cokernel of theinjectionOX(D) OX(D)is flat overS.

Proof. By Proposition2.1.5we have a relative effective Cartier divisor D

=D D; let : D Xbe the embedding. Tensoring the exact sequence

0 ID OX OD 0

with OX(D)yields the exact sequence

0 OX(D) OX(D) OX(D) OD 0

because

ID OX(D) = OX(D) O(D) = OX(D D

) = OX(D).

Hence the cokernel ofOX(D) OX(D)is given by OX(D) OD, whichis flat because it is the tensor product of flat sheaves.

Proposition 2.1.7. LetR be a Noetherian ring, letS= Spec(R), letX/Sbe a relative curve and let F be an OX-module which is flat over S. IfH1(X,F)is a projectiveR-module, then so isH0(X,F).

Proof. Let U = {Ui}iIbe an open affine covering of Xwhich is closedunder intersection. Since X is Noetherian we may assume that I is finite,sayI= {1, . . . , n}. WriteUij =Ui Uj. Consider the map of ech modules

d0 :n

i=1

H0(Ui,F)

1i


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Now, since H1(X,F)is projective, we obtain an exact sequence

0 Im(d0)

1i


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2.2 Criteria for very ampleness 21

2.2 Criteria for very ampleness

Let f: X Ybe a morphism of schemes and let Fbe a coherent sheaf onX. Then the ith higher direct imageofFis the sheafRif(F) associatedto the presheaf

V Hi(f1(V),F|f1(V))

onY. The most important property ofRif(F)for our purposes is given bythe following proposition.

Proposition 2.2.1.LetXbe a Noetherian scheme, letY= Spec(A)be anaffine scheme, and letf: X Ybe a morphism. Then for any quasi-coherent

OX-moduleF onX, we have

Rif(F) =Hi(X,F)

whereHi(X,F) denotes the sheaf onYassociated to theA-moduleHi(X,F).

Proof. See Hartshorne [17,Chapter III, Proposition 8.5].

Theorem 2.2.2. Let f :X S be a projective morphism of Noetherianschemes and letFbe a coherent sheaf onXwhich is flat overS. Letsbe apoint inS.

(i) If the natural map

(s)i : Rif(F) k(s) Hi(Xs,Fs)

is surjective, then it is an isomorphism in a neighbourhood ofs.

(ii) If(s)i is surjective, then the following conditions are equivalent.

(a) (s)i1 is surjective;

(b) Rif(F)is locally free in a neighbourhood ofs.

Proof. See Hartshorne [17,Chapter III, Theorem 12.11].

Proposition 2.2.3.For0< i < nand alld Zwe have

Hi(PnS,OPnS(d)) = 0.

Proof. See Hartshorne [17,Chapter III, Theorem 5.1(b)].

The following two lemmas give convenient reformulations of NakayamasLemma (see Atiyah and Macdonald[1, Proposition 2.6]).


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Lemma 2.2.4. LetAbe a ring and letMbe a finitely generatedA-module.

IfM/mM= 0for all maximal idealsmofA, thenM= 0.

Proof. We have M = 0 if and only ifMm = 0 for all maximal ideals m ofA, so we reduce to the case where A is local. Then m coincides with theJacobson radical ofAand so the result follows by Nakayamas Lemma.

Lemma 2.2.5. Let A be a ring, let u :M N be a homomorphism ofA-modules, and assume that N is finitely generated. If the induced homo-morphismsum : M/mM N/mNare surjective for all maximal idealsmofA, thenuis surjective.

Proof. By assumption N/u(M) A/m = 0 for all maximal ideals m A,hence u(M) =Nby Lemma2.2.4.

Lemma 2.2.6. For alld 1we have

H1(P1S,OP1S(d)) = 0.

Proof. Let s be a closed point of S and let k = k(s). Then by Theo-rem1.2.4(ii), we have

H1(P1k,OP1k(d)) =H0(P1k, P1k OP1k(d)) = 0

where P1k is the canonical sheaf on P1k, and so the natural map

:H1(P1S,OP1S (d)) H1(P1k,OP1k(d))

is (trivially) surjective. Then Theorem2.2.2(i) implies that is an isomor-phism and so H1(P1S,OP1S (1)) k(s) = 0 for all closed s. The result thenfollows from Lemma2.2.4.

Proposition 2.2.7.LetX/Sbe a relative curve and letLbe a very ampleinvertible sheaf onX/S. ThenH1(X,L) = 0.

Proof. As L is very ample we have L =OPn(1)for some n 1. Then

H1(X,L) =H1(Pn, OPn(1)) =H

1(Pn,OPn(1)) = 0

where the last equality follows from Proposition2.2.3when n 2from andLemma2.2.6when n= 1.

Proposition 2.2.8. Let f :X S be a relative curve over S = Spec(R)and letLbe a very ample invertible sheaf onX. Then for all closed pointss S, we have

H0(X,L) Rk(s) =H0(Xs,Ls).


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Proof. Let s be a closed point in S. We have H1(X,L) = 0 by Propo-

sition 2.2.7 so R1f(L) and H1(Xs,Ls) are zero. Hence (s)1 is (triv-ially) surjective and R1f(L) is (trivially) locally free at s. Thus (s)0 :R0f(L) k(s) H0(Xs,Ls)is surjective by Theorem2.2.2(ii) and there-fore an isomorphism by Theorem2.2.2(i). Taking global sections we see thatH0(X,L) k(s) =H0(Xs,Ls)as required.

Let Lbe a locally free coherent sheaf on a Noetherian scheme X. LetSymd(L) denote the dth symmetric tensor product of L and let S =

d0Symd(L)be the symmetric algebra ofL. It is a sheaf ofOX-algebras.

Theprojective space bundleofLis defined to be the global ProjofS, andis denoted by P(L).

Proposition 2.2.9. Letf: X Sbe a quasi-compact morphism and letLbe an invertible sheaf onX. ThenLis very ample if and only if

(i) the canonical map ffL L is surjective, and

(ii) the induced map X P(fL)is an immersion.

Proof. See Grothendieck[14, Proposition 4.4.4]

Corollary 2.2.10. Letf: X Sbe a quasi-compact morphism of schemesand letLbe an invertible sheaf onX. ThenL is very ample if and onlyif there exists an open covering{U}ofSsuch thatL|f1(U) is very amplerelative to U for all.

Proof. See Grothendieck[14, Corollaire 4.4.5].

Proposition 2.2.11. LetX/Sbe a scheme and letL andL be very ampleOX-modules. ThenLL is very ample.

Proof. See Grothendieck[14, Corollaire 4.4.9(ii)].

Proposition 2.2.12. Letf :X Sbe a relative curve, letLbe an invert-

ible sheaf onX. ThenL

is very ample if and only ifL

s is very ample forall closed pointss S.

Proof. SupposeL is very ample and let s Sbe a closed point. By definitionL induces a closed immersion :X PnSsuch that L =

OPnS

(1). Veryampleness is local on Sby Corollary2.2.10, so Ls = ( idk(s))OPn

k(s)(1)

gives a closed immersion and hence Lsis very ample.To see the converse, we check the conditions of Proposition2.2.9, noting

that they hold on each closed fibre. Since fs (fs)Ls Ls is surjective,it is surjective on stalks. Hence ffL L is surjective on stalks by


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2.3 Tensor products and sheaf quotients 24

Nakayamas Lemma and is thus surjective. Secondly,iLs : Xs P((fs)Ls)is

a closed immersion for each closeds Sand H0(X,L) k(s) =H0(Xs,Ls)follows from Proposition2.2.8. Hence P((fs)Ls) = P(fL) Spec(k(s))forall closed sand so iL :X P(fL)is a closed immersion.

Corollary 2.2.13. LetX/Sbe a relative curve of genusg and letL be aninvertible sheaf onX. Ifdeg(L) 2g+ 1, thenL is very ample.

Proof. The lower bound on the degree implies that Lsis very ample for allclosed points s Sby Riemann-Roch. Hence Lis very ample by Proposi-tion2.2.12.

2.3 Tensor products and sheaf quotients

In this section we prove Propositions 2.3.5 and2.3.7, which form the ba-sis of all the algorithms for performing divisor arithmetic described in thenext chapter. These two propositions are generalisations of Khuri-Makdisi[22, Lemmas 2.2 and 2.3] to the case of relative curves. Given a very amplesheafLon a relative curve Xover Spec(R), these two propositions amountto a description of the R-algebra structure of the homogeneous coordinatering

i1 H

0(X,Li) ofXwith respect to the embedding by L. Proposi-tions2.3.5will be realised explicitly in Algorithm3.4.1and Proposition2.3.7in Algorithm3.4.2.

Recall that an ample invertible sheafLon a relative curve X/S isnor-mally generatedif the maps

H0(X,L)d H0(X,Ld)

are surjective for all d 1.The following proposition is a direct generalisation of a proof of Mumford

[31, pp3839].

Proposition 2.3.1.LetX/Sbe a relative curve and letLbe an invertible

sheaf onX. IfLis normally generated, then it is very ample.

Proof. Let Lbe normally generated. As Lis ample we obtain morphisms

L: X Pm and Ld : X P

n

for all d 1. The d-uple embedding ofPm is a morphism vd: Pm Pm

where m =

m+dm

1. We can identify Pn with a subspace ofPm

via thesurjections

H0(X,L)d H0(X,Ld).


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We thus obtain the following commutative diagram:

XLd

L

Pn

Pm vd Pm

As L is ample, Ld is very ample for all sufficiently large dand hence Ldis a closed immersion for all sufficiently large d. Hence, from the diagram wesee that Lis also a closed immersion and is thus very ample.

Proposition 2.3.2.LetX/Sbe a relative curve and letLbe an invertible

sheaf onX. ThenL is normally generated if and only ifL is very ampleand the natural maps

H0(Pn,OPn(d)) H0(X,Ld)

are surjective for alld 1.

Proof. IfLis normally generated, then it is very ample by Proposition2.3.1.For alld 1, we have canonical isomorphisms

H0(Pn,OPn(d)) =H0(Pn,OPn(1))

d =H0(X,L)d.

Hence, for alld 1, we see thatH0(X,L)d H0(X,Ld)is surjective if

and only ifH0(Pn,OPn(d)) H0(X,Ld)is surjective.

Proposition 2.3.3. LetX/Sbe a relative curve of genusgand letLbe aninvertible sheaf onX. Ifdeg(L) 2g+ 1, thenL is normally generated.

Proof. The lower bound on the degree implies Lis very ample by Proposi-tion2.2.13. Then by Proposition2.3.2it remains to prove that

L :H0(Pn,OPn(d)) H

0(X,Ld)

is surjective for alld 1. For any closed pointsofSwe have a commutativediagram

H0(Pn,O(d))

L

H0(X,Ld)

H0(Pnk(s),O(d)s) Ls

H0(Xs,Ld

s)

where the mapLs

is surjective by Proposition2.3.2because Lsis normallygenerated by Corollary1.3.9, and the vertical maps arise from taking tensorproducts with k(s). As this holds for all closed points, we see that L issurjective by Lemma2.2.5.


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Lemma 2.3.4. The natural map

H0(PmS,OPmS (1)) H0(PnS,OPnS(1)) H

0(PmS PnS,OPmSPnS (1))

is surjective.

Proof. Letp1 :Pm Pn Pm andp2 :Pm Pn Pn be the projection maps.We have

p1OPmS (1) p2OPnS

(1) = OPmSPn

S(1)

(see Hartshorne [17,Chapter II, Exercise 5.2]), so it suffices to prove that

H0(PmS,OPmS (1)) H0(PnS,OPnS(1)) H

0(PmS PnS, p

1OPmS

(1) p2OPnS(1))(2.1)

is surjective. Let s be a closed point of Sand let k = k(s). Then Theo-rem1.2.5implies that the map

H0(Pmk,OPmk (1)) H0(Pnk ,OPnk (1)) H

0(Pmk Pnk , p

1OPmk

(1) p2OPnk (1)),

obtained by tensoring (2.1) withk , is an isomorphism. Hence (2.1) is surjec-tive by Lemma2.2.5.

Proposition 2.3.5.LetX/Sbe a relative curve and letM andN be nor-mally generated sheaves onX/S. Then

MN : H0(X,M) H0(X,N) H0(X,MN) (2.2)

is surjective.

Proof. By Proposition2.3.2there exist integers m and n such that the maps

H0(Pm,OPm(1)) H0(X,M) and H0(Pn,OPn(1)) H0(X,N)

are surjective. Now M N is very ample by Proposition 2.2.11and theembedding it defines factors through the Segre embedding, hence the map

H0(Pm Pn,OPmPn(1)) H0(X,MN)

is surjective. But

H0(Pm,OPm(1)) H0(Pn,OPn(1)) H

0(Pm Pn,OPmPn(1))

is surjective by Lemma2.3.4,and so we obtain a commutative diagram

H0(Pm,OPm(1)) H0(Pn,OPn(1))

H0(X,M) H0(X,N)

0

H0(Pm Pn,OPmPn(1)) H0(X,MN) 0

where all maps exceptare known to be surjective. Thus is surjective.


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Combining Propositions2.3.3and2.3.5we see that, ifM and L have

sufficiently large degrees, then H0(X,M) and H0(X,N) contain all theinformation needed to describe H0(X,M N), which in turn completelydescribes MN.

For any submodules N N and P P, define the module quotientof P by N with respect to a homomorphism : M N P to be thesubmodule

(P :N)= {m M | (m N) P}

ofM. (Similarly, one can define (P :M) Nfor a submodule M ofM,but we will not need this.) We will often drop from the notation whenit clear from the context. By definition the module quotient satisfies thefollowing property: for any submodule M M, if(M N) P, thenM (P :N).

Proposition 2.3.6. LetX/Sbe a relative curve of genus g and lets be aclosed point ofS. LetMs andNs be invertible sheaves onXs and assumeNs is generated by global sections. Then for any divisorDonXs, we have

H0(Xs,Ms(D)) =

H0(Xs,Ms Ns(D)) :H0(Xs,Ns)

MsNs

with respect to the canonical map

MsNs : H0(Xs,Ms) H

0(Xs,Ns) H0(Xs,Ms Ns).

Proof. See Khuri-Makdisi [22,Lemma 2.3].

Proposition 2.3.7.LetX/Sbe a relative curve of genusg and letM andN be invertible sheaves onX, each of degree at least2g+ 1. Then for anyrelative effective Cartier divisorD onXof degree at mostdeg(M)(2g +1),we have

H0(X,M(D)) =

H0(X,MOX N(D)) :H

0(X,N)

MNwith respect to the canonical homomorphismMN of (2.2).Proof. Let sbe a closed point ofS. Let

M=H0(X,M), N=H0(X,N), P=H0(X,MN)

M =H0(X,M(D)), and P =H0(X,MN(D)).

Note thatP=H0(X,M(D)N). Denote the corresponding modules onthe fibre ofXover sby a subscript s; for example Ms=H

0(Xs,M(D)s).


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We have a homomorphism : M N P obtained from : M NPby restriction. This induces a homomorphism which, after applying Propo-sition2.2.8,is given by : Ms Ns P

s. Then Proposition2.3.6gives the

equalityMs= (P

s:Ns). (2.3)

Now, by the definition of the module quotient we have

M (P :N). (2.4)

Applying Proposition2.2.8, we obtain an induced homomorphism

(P :N) k(s) Ns Ps,and so by (2.3) we have

(P :N) k(s) (Ps: Ns) =Ms. (2.5)

Hence, combining (2.4) with (2.5), we obtain (P :N)/M k(s) = 0for allclosed pointss Sand so (P :N) =M by Lemma2.2.4.


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Chapter 3

Divisor arithmetic on relative

JacobiansIn this chapter we describe how to perform arithmetic of relative effectiveCartier divisors on a relative curve. The choice of base ring is limited onlyby the presence of effective linear algebra functions; we formalise this notionin Section 3.2. In Section 3.3 to describe how to calculate homomorphicimages of modules and module quotients with respect to a given bilinearmap. This leads to the algorithmic realisation of Propositions2.3.5and2.3.7in Section3.4. Finally, these algorithms are used to describe arithmetic ofdivisors on relative curves in Section3.5, and on relative Jacobians of relativecurves in Section3.6, in a manner directly analogous to the algorithms ofKhuri-Makdisi[22].

3.1 Complexity analysis

LetA = R[x1, . . . , xs]be a multivariate polynomial ring with real coefficientsand let fbe an element of A. The big O class off, denoted by O(f), isdefined to be the set consisting of the polynomials g Awhich satisfy the

following condition: there exist Mg, Cg R such that |g(m1, . . . , ms)| Cg|f(m1, . . . , ms)|for all (m1, . . . , ms) Rs satisfying mi Mg for all i =1, . . . , s.

LetMbe a projective R-module. ThenMis locally free and so there is amaprankM: Spec(R) N defined byrankM(p) = rank(Mp)whererank(Mp)is the rank of the free Rp-module Mp. The map rankM is locally constanton Spec(R), and, since R is Noetherian, takes on finitely many values onX.We define thesizeofMto be max{rankM(p) | p Spec(R)}. The exampleto keep in mind is of a free module Rm where the size is simply m.


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3.2 Amenable rings 30

Thespace complexityof an R-module is an estimation of the amount of

storage space required to represent it. In the following sections, the spacecomplexity will be estimated relative to the sizes of certain fixed projectivemodules.

By the time complexityof a procedure, we mean the asymptotic num-ber of arithmetic operations in the base ring R required by the procedureto execute, where an operation in the base ring R is an addition, subtrac-tion, multiplication, division by a unit and assignment of elements ofR. Wewill always assume that multiplications and divisions by units are asymp-totically more expensive than additions, subtractions and assignments. Theperformance of the algorithms we discuss will be expressed in terms of the

time complexity of the steps that constitute them as a function of the spacecomplexity of the input.LetMbe a projectiveR-module of sizem. We will assume that the space

complexity ofMis inO(m). IfNis a second projective R-module of sizen,we will assume that the space complexity of a homomorphism : M N isin O(mn). This is analogous to the case of a homomorphism : Rn Rm

between free modules being represented by an m nmatrix.A finitely generated submodule Nof a projective module Mof size m

will always be given as a homomorphic image; that is Im() = N for somehomomorphism :Rn M. We will often write simply N M leaving

the homomorphism defining N implicit. By rights, the space complexityof the submodule Nis the space complexity of its defining homomorphism,which is in O(mn). However, we will make a further assumption that thespace complexity ofNis in O(m2). This assumption will always hold in theapplications in Sections3.4,3.5and3.6 and serves to considerably simplifythe expressions describing the time complexities of the algorithms.

3.2 Amenable rings

There are several operations that we need to perform projective R-modules,

their finitely generated submodules, and homomorphisms between them.Here we give a name to those rings for which the module operations canbe computed. We will say that a ring R isamenableif we can perform ex-act arithmetic on elements ofR, and the following functions are effectivelycomputable on projective R-modules and homomorphisms between them:

Dual: Given a homomorphism : M N, return the dual homomorphismbetween the dual modules : N M where M = Hom(M, R). Ifthe space complexities ofMandNare inO(m)and O(n)respectively,then we assume that the space complexities ofM, N and are in


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3.2 Amenable rings 31

O(m), O(n)and O(mn)respectively, and that the computation of

has time complexity inO(mn).

Composite: Given homomorphisms : M N and : N P, returnthe homomorphism : M P. If the space complexities ofM, Nand Pare in O(m),O(n), and O(p)respectively, then we assume thatthe space complexity of is in O(mp)and that the computation of has time complexity in O(mnp).

Kernel: Given a homomorphism : M N, return a homomorphism : K M for some K such that Im() = Ker(). If the spacecomplexities ofMand Nare in O(m)and O(n)respectively, then weassume thatKhas space complexity inO(m), so has space complex-ity in O(m2). We denote the time complexity to compute Ker() byK(m, n).

Common kernel: Given homomorphisms i : M N for i = 1, . . . , r,return the common kernel

ri=1Ker(i) of the i. If the space com-

plexities of M and N are in O(m) and O(n) respectively, then weassume that the space complexity of the common kernel is in O(m2).We denote the cost of computing the common kernel by CK(r,m,n).

Sum: Given submodulesM1 MandM2 M, return the sumM1 + M2M. If the space complexity ofM is in O(m), then the space complex-ities of M1 and M2 will be in O(m2) and we assume that the spacecomplexity ofM1 + M2is in O(m2). We denote the time complexity ofcomputing M1+ M2by S(m).

Note that the assumptions for the number operations of each of the func-tions above are inspired by the analogous case of homomorphisms betweenfree modules given by matrices. Of course the correctness of the followingalgorithms is not effected in case these assumptions fail to hold for a givenamenable ring.

As the primary application is to finite rings, we have made no attempt toanalyse the size of intermediate expressions of the algorithms, although thiscan certainly play a significant role for calculations over infinite rings. Whereapplicable, working in a quotient ring or employing appropriate variants ofthe LLL algorithm are common techniques for controlling intermediate ex-pression swell. See Cohen [9,Chapter 2] for some examples over the integers.

For the remainder of this section we provide a brief survey of some ringswhich are known to be amenable. Cohen [9] discusses the two classicalamenable rings, namely fields with exact arithmetic and the ring of integers


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3.3 Arithmetic of modules 32

(or indeed any domain with exact arithmetic and an effectively computable

Euclidean function).Recall that a principal ideal ring is ring in which every ideal is principal;

such a ring need not be a domain in general. The work of Storjohann [37],Mulders and Storjohann[29] and Buchmann and Neis [5]shows that a largeclass of principal ideal rings and their quotients are amenable. This classincludes all finite semi-local rings, and in particular, quotients of discretevaluation rings.

The work of Bosma and Pohst [3], Cohen [10] and Neis [33]shows thatDedekind domains are amenable. Caruso and Lubicz [6] have announcedalgorithms that make certain quotients of the ring Zp[[u]]amenable.

3.3 Arithmetic of modules

In this section we describe the algorithms that allow us to evaluate a givenhomomorphism : M N Pbetween projective modules M, N and P,and to compute the module quotient (also known as the colon module; notto be confused with a quotient module) with respect to .

The following operations are described for vector spaces by Khuri-Makdisi[23, Definition 2.5] and algorithms are given by [23, Proposition 2.6]. (Notethat Khuri-Makdisi provides several different representations of in [23,

Section 2]; we follow what he calls Representation A.) We will describethe analogous operations and algorithms for finitely generated submodulesof fixed projective R-modules.

Throughout the remainder of this section we suppose given fixed projec-tive R-modules M, Nand P, and a homomorphism

: MRN P.

Any element xin Minduces a homomorphism

x : N P defined by x(y) =(x y).

Similarly, we obtain a homomorphism y : M P for any element y inN.If{xi| i= 1, . . . , m}, {yj | j = 1, . . . , n} and {zk | k= 1, . . . , p} are fixed

generating sets forM,Nand Prespectively, then can be given as follows:For alli = 1, . . . , mand j = 1, . . . , n, there exist elements aijk Rsuch that

(xi yj) =

k

aijk zk.

The (j, k)entry of the underlying matrix ofxi will then be aikj . Similarlythe (i, k)entry of the underlying matrix ofyj will be akji .


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3.3 Arithmetic of modules 34

Proof. In step2, we compute (gi N) P for each i = 1, . . . , m. Then

step3calculatesm

i=1

(gi N) =(M N).

There are m O(m) calls to Algorithm3.3.1in step2; hence the timecomplexity of step2is in O(mnp(m+ n)). The space complexity of eachsubmodule (gi N) is in O(p2), as is the sum of two such modules, sothe time complexity to calculate the sum in step 3is in O(mS(p)) and thespace complexity of the result is inO(p2). The time complexity of the wholealgorithm is thus in O(mnp(m + n) + mS(p)).

The following algorithm computes the module quotient (P : N) ofsubmodules P P and N N. It is analogous to that of Khuri-Makdisi[23, Algorithm 2.6(4)].

Algorithm 3.3.3 (Module quotient). Given finitely generated projectivesubmodules N N and P P where P is a direct summand ofP, thefollowing algorithm calculates the submodule (P : N) ofM. If the spacecomplexities ofN andP are inO(n2)and O(p2)respectively, then the timecomplexity of the algorithm is in O(mnp(n +p) + K(p, p) + CK(n,m,p))andthe space complexity of(P :N)is in O(m2).

1. Let : Rp Pbe such that Im() =P and p O(p).

2. Form the dual : P Rp

of.

3. Compute the kernel : Rk P of.

4. Take the dual ofto obtain : P Rk.

5. Let{g1, . . . , gn}be a generating set for N where n O(n).

6. For each i= 1, . . . , n, compute the multiplication-by-gihomomorphismg

i

: M P.

7. For each i= 1, . . . , n, compute the composite gi : M Rk.

8. Return the intersection of the kernelsn

i=1Ker( gi).

Proof. Let : K Mbe the intersection calculated in step8, that is

(K) =n

i=1

Ker( gi).


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We need to show that (K) = (P : N). First, steps2and3construct a

sequenceRk P

Rp

(3.1)

which is exact at P. We first show that Ker() =P. Since P is a directsummand of P, we have P = P P for P = P/P projective. Thenexactness of (3.1) implies Im() = Ker() = (P/P). Equating P andP, we see that

Ker() = {x P | (x) = 0for all (P/P)}.

But (P/P) is a direct summand ofP because direct sums commute withduals, hence if x = x +x for x P and x P, then (x +x) =(x +P) and since x +P is nonzero there exists a function in (P/P)

which is nonzero at x +P. Hence Ker() = P. The homomorphism

is calculated in step4.Now let v (K). Then v

ni=1Ker(

gi) if and only if v Ker( gi)for all iif and only if(v gi) Ker(

) =P for all iif andonly if(v N) P. Hence (K) = (P :N)as required.

The time complexity to compute the dual of in step2 is in O(p2), tocompute the kernel in step3is in O(K(p, p))since k O(p)by assumption,and to compute the dual in step4is inO(p2). The total time complexity for

steps2to 4is thus in O(p2 + K(p, p)), and the space complexity of is inO(p2).

The time complexity to calculate each giin step6is inO(mnp)and eachhas space complexity in O(mp). Since there are n O(n)homomorphismsgi , the time complexity of step6is in O(mn

2p).For each i = 1, . . . , n, the time complexity to calculate each composite

gi in step7 is in O(mp2) since the space complexities ofgi: M P

and : P Rk are in O(mp)and O(p2) respectively. There are n O(n)such composites, so the time complexity of step 7is in O(mnp2).

In step8the time complexity to compute the common kernel is CK(n,m,p)

sincen

O(n)and the space complexity of each

giis inO(mp). Hencethe total time complexity of the algorithm is in O(mnp(n+p) + K(p, p) +CK(n,m,p)). The space complexity of (P : N) M is in O(m2) by as-sumption.

3.4 Tensor products and sheaf quotients

Recall from Theorem1.2.3that, for any invertible sheafL on X, the setH0(X,L)of global sections ofLis a finitely generated R-module. We will


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represent relative effective Cartier divisors on a relative curveXby first fixing

a very ample invertible sheafLwith large degree whose module H0(X,L)of global sections is projective. Then a relative effective Cartier divisor D onXwill be given by a set of generators for the finitely generated R-submoduleH0(X,L(D)). By choosing L such that L(D) is very ample, D canalways be recovered from H0(X,L(D)), at least in principle.

The algorithms in Section3.3allow us to explicitly compute the resultsof Propositions2.3.5and2.3.7. Let R be an amenable ring, letS= Spec(R)and let X/Sbe a relative curve. Let Lbe a normally generated invertiblesheaf on X. Then Propositions2.1.7and2.3.5imply that

ij : H0

(X,Li

) H0

(X,Lj

) H0

(X,Li+j

)

is a surjective homomorphism of projective R-modules.We assume that the size of H0(X,Li) is in O(ig) for all i 1. In

practice, the maximum i required is determined by the maximum degreerelative effective Cartier divisor one wishes to represent. Thus in the case ofarithmetic on Pic0X(S), as we consider in Section3.6,iis bounded above bya small constant (in Algorithms3.6.3and3.6.4we have i 5).

Algorithm 3.4.1 (Product of global sections). Let i and j be positive in-tegers, let Mbe a subsheaf ofLi and let Nbe a subsheaf ofLj, each of

degree at least 2g+ 1. Given the submodules H0(X,M) H0(X,Li)andH0(X,N) H0(X,Lj), the following procedure calculates the submoduleH0(X,MN)ofH0(X,Li+j). If the space complexities ofH0(X,M)andH0(X,N)are inO(i2g2)and O(j2g2)respectively, then the time complexityof the algorithm is in O(ij(i +j)2g4 + igS((i +j)g))and the space complexityof the result is in O((i +j)2g2).

1. Return H0(X,MN) obtained by applying Algorithm3.3.2to themodulesH0(X,M)and H0(X,N).

Proof. By the hypothesis on the lower bounds of the degrees ofM and

N, Proposition2.3.5implies that H0(X,M) H0(X,N) surjects via ijonto H0(X,MN). The call to Algorithm3.3.2calculates this image ofH0(X,M) H0(X,N). The time complexity of the call to Algorithm3.3.2is inO(ij(i+j)2g4+igS((i+j)g))and the space complexity ofH0(X,MN)is in O((i+j)2g2)since it is a submodule ofH0(X,Li+j)which has spacecomplexity in O((i +j)g)by assumption.

The following algorithm is a generalisation of the descriptions providedby Khuri-Makdisi in[22, Remark 3.8] and[23, Proposition 2.6(4)].


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3.5 Arithmetic of divisors 37

Algorithm 3.4.2(Sheaf quotient). Let iand j be positive integers, let M

be a subsheaf ofLi and let Nbe a subsheaf ofLj, each of degree at least2g + 1. Let Ebe a relative effective Cartier divisor on X whose degreesatisfies

deg(E) min{deg(M), deg(N)} (2g+ 1).

Given the submodules H0(X,M) H0(X,Li), H0(X,N) H0(X,Lj)and H0(X,N(E)) H0(X,Lj), the following procedure calculates thesubmodule H0(X,M(E)) ofH0(X,Li). If the space complexities of thesubmodules H0(X,M),H0(X,N)and H0(X,N(E))are in O(i2g2),O(j2g2)and O(j2g2) respectively, then the time complexity of the algorithm is inO(ij(i +j)2g4 + igS((i +j)g) +K((i +j)g, (i +j)g) +CK(jg, ig, (i +j)g))andthe result has space complexity in O(i2g2).

1. Apply Algorithm3.4.1 to H0(X,M) and H0(X,N(E)) to obtainthe module H0(X,MN(E)).

2. Return the module H0(X,M(E))obtained by applying Algorithm3.3.3toH0(X,MN(E))and H0(X,N).

Proof. By Proposition 2.3.5, the result of step 1 is H0(X,M N(E))which is isomorphic to H0(X,M(E) N). The bounds on the degreesimply that all the sheaves under consideration are normally generated by

Proposition2.3.3thus have zero higher cohomology by Proposition2.2.7andare hence projective by Proposition2.1.7. ThenH0(X,MN)/H0(X,MN(E)) is projective by Corollary2.1.8and so H0(X,MN(E)) is adirect summand ofH0(X,MN). This proves that the arguments satisfythe conditions for the application of Algorithm3.3.3in step2. Then Propo-sition2.3.7implies that the module quotient calculation in step2gives

H0(X,MN(E)) :H0(X,N)

=H0(X,M(E))

as required. Step1has time complexity in O(ij(i+ j)2g4 +igS((i+ j)g))and step2has time complexity in O(ij(i+ j)2g4 + K((i+ j)g, (i+ j)g) +

CK(jg, ig, (i+j)g)). Hence the time complexity of the algorithm is in O(ij(i+j)2g4 + igS((i +j)g) + K((i +j)g, (i +j)g) + CK(jg, ig, (i +j)g)). The resultH0(X,M(E))is a submodule ofH0(X,Li)hence has space complexity inO(i2g2)by assumption.

3.5 Arithmetic of divisors

In this section we give descriptions of the arithmetic operations on divisorsgiven as modules of global sections as described in the previous section.


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3.5 Arithmetic of divisors 38

To this end we employ the theory developed in Section3.3to demonstrate

that the main algorithms of Khuri-Makdisi[22] continue to hold for relativeeffective Cartier divisors on relative curves overS= Spec(R)for an amenableringR.

Throughout this section and the next we fix the following notation. LetR be an amenable ring, let S = Spec(R) and let X/Sbe a relative curveof genus g. Let Lbe an invertible sheaf on X of degree 2g+ 1, soL is normally generated by Proposition2.3.3. We will assume the size ofH0(X,Li) is in O(ig) for all i 1 and that the space complexity of asubmodule ofH0(X,Li)is in O(i2g2).

The first algorithm is analogous to Khuri-Makdisi [22, Algorithm 3.6].

Algorithm 3.5.1 (Divisor addition). Let D1 and D2 be relative effectiveCartier divisors on X satisfying deg(D1+ D2) (2g+ 1). Given thesubmodules H0(X,L(D1))and H0(X,L(D2))ofH0(X,L), the follow-ing procedure calculatesH0(X,L(D1 D2)). If the space complexities ofH0(X,L(D1))and H0(X,L(D2))are in O(g2), then the time complex-ity of the algorithm is in O(g4 + gS(g) + K(g, g) + CK(g , g , g))and the resulthas space complexityO(g2).

1. Apply Algorithm3.4.1toH0(X,L(D1))andH0(X,L(D2))to ob-tain the submodule H0(X,L2(D1 D2))in H0(X,L2).

2. ReturnH0(X,L(D1 D2))obtained by applying Algorithm3.4.2tothe modulesH0(X,L),H0(X,L2)and H0(X,L2(D1 D2)).

Proof. Step1produces H0(X,L2(D1 D2)) since deg(D1+ D2) (2g+ 1)implies deg(L(Di)) 2g+ 1for i= 1, 2. Now deg(L) 2g+ 1and deg(L2(D1 D2)) 2g+ 1, so the application of Algorithm3.4.2instep2returns H0(X,L(D1 D2)).

The time complexity of step1is inO(g4+gS(g))and the time complexityof step2is inO(g4 +gS(g) +K(g, g) +CK(g , g , g)), hence the time complexityof the algorithm is in O(g4 + gS(g) + K(g, g) + CK(g , g , g))operations.

The following algorithm is analogous to Khuri-Makdisi [22,Algorithm 3.10].

Algorithm 3.5.2(Divisor flip).LetDbe a relative effective Cartier divisoron X of degree at most (2g+ 1) and let s be a non-zero element ofH0(X,L(D)). Considering sas a section ofH0(X,L), write

div(s) =E=D+ D

where H0(X,OX(D)) =H0(X,L(D)). Given the submodule H0(X,L(D))ofH0(X,L)and the element sofH0(X,L(D)), the following procedure


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3.6 Arithmetic on a relative Jacobian 39

computes H0(X,L(D)). If the space complexity ofH0(X,L(D))is inO(g2), then the time complexity of the algorithm is in O(g4+gS(g)+K(g, g)+CK(g , g , g))and the space complexity of the result is in O(g2).

1. Compute H0(X,L2(D D))by applying Algorithm3.4.1to sandH0(X,L).

2. Return H0(X,L(D)) obtained by applying Algorithm3.4.2to themodulesH0(X,L),H0(X,L(D))and H0(X,L2(D D)).

Proof. SincesgeneratesH0(X,L(D D))and sincedeg(L(D D)) =deg(L) 2g+ 1, Algorithm3.4.1returns H0(X,L2(D D)) in step1.

As deg(L(D)) 2g+ 1, the call to Algorithm 3.4.2in step2producesH0(X,L(D))as claimed.The time complexity of the call to Algorithm3.4.1in step1is in O(g4 +

gS(g))operations. The time complexity of step2is inO(g4+gS(g)+K(g, g)+CK(g , g , g)). Hence the time complexity for the algorithm is in O(g4+gS(g)+K(g, g) +CK(g , g , g)). The result is a submodule ofH0(X,L)and hence hasspace complexity in O(g2).

3.6 Arithmetic on a relative Jacobian

Khuri-Makdisi[22] describes three models for divisor class arithmetic in thePicard group of a curve: the large, medium and small models. Thoughthe complexity of the algorithms for each of the three models is the same,the medium and small models reduce the size of the vector spaces used bya constant factor at the expense of more complicated algorithms. We willdescribe the algorithms in terms of the medium model. We keep the notationas described at the beginning of the previous section.

Recall that, for a scheme X, the Picard group, Pic(X), is the groupH1(X,OX) of isomorphism classes of invertible sheaves on Yand that themap X Pic(X) is a functor. Letf: X Sbe a relative curve and let

T Sbe an S-scheme. For the purposes of this section, we definePic0X(T) = {L Pic(XT) | deg(Lt) = 0for allt T}/f

TPic(T).

WhenX/Sis a relative curve, the groupPic0X(S)can be given the structure ofa scheme; it thus becomes an abelian S-scheme whose fibres are isomorphicto the Jacobians of the fibres ofX. For more details, see Bosch et al. [2,Chapter 9, Section 4].

Let Mbe an invertible sheaf of degree 0. Then the isomorphism classofM is represented by any relative effective Cartier divisor D of degree


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deg(L) such that M = L(D). Since deg(L) 2g+ 1, for such D we

havedeg(L2(D)) = deg(L) 2g + 1and so Proposition2.3.3implies thatL2(D)is normally generated. Elements ofPic0X(S)will be represented inthis way, as the submodule H0(X,L2(D))ofH0(X,L2)for some relativeeffective Cartier divisor Dof degree deg(L).

Algorithm 3.6.1 (Zero element). The following procedure computes thezero element ofPic0X(S). The time complexity of the algorithm is in O(g

4 +gS(g))and the space complexity of the result is in O(g2).

1. Choose any non-zero element sofH0(X,L).

2. Apply Algorithm3.4.1to sand H0(X,L)and return the result.Proof. This algorithm constructs a submodule corresponding to the relativeeffective Cartier divisorD = div(s), which is in the zero class inPic0X(S). Instep2,the result of Algorithm3.4.1is (s H0(X,L)) =H0(X,L2(D)).The time complexity is determined by the call to Algorithm3.4.1,hence is inO(g4 + gS(g)). The result is a submodule ofH0(X,L2)and hence its spacecomplexity is in O(g2).

The following algorithm is analogous to that of Bruin [4, Algorithm 2.10].

Algorithm 3.6.2 (Zero test). Let xbe a point in Pic0

X(S)given by a sub-moduleH0(X,L2(D))ofH0(X,L2). The following procedure determineswhether x = 0. If the space complexity ofH0(X,L2(D))is in O(g2), thenthe time complexity of the algorithm is inO(g4+gS(g)+K(g, g)+CK(g , g , g)).

1. Apply Algorithm3.4.2to H0(X,L),H0(X,L2)and H0(X,L2(D))to obtain H0(X,L(D)).

2. ReturnfalseifH0(X,L(D)) = 0and trueotherwise.

Proof. Correctness follows from the fact that L2(D)is trivial if and only ifthere exists an elements H0(X,L(D))such thatD = div(s). This is thetest performed in step2. The calculation is dominated by Algorithm3.4.2and so the time complexity is in O(g4 + gS(g) + K(g, g) + CK(g , g , g)).

The following algorithm is analogous to that of Khuri-Makdisi [22, Algo-rithm 5.1]; see also Bruin [4,Algorithm 2.11].

Algorithm 3.6.3 (Addflip). Let x and y be elements ofPic0X(S)given bysubmodules H0(X,L2(D1)) and H0(X,L2(D2)) of H0(X,L2). The


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following procedure computes the submodule H0(X,L2(E))and a global

section sofH0(X,L3), where sand Esatisfy

div(s) =D1+ D2+ E.

If the space complexities ofH0(X,L2(D1)) and H0(X,L2(D2)) are inO(g2), then the time complexity of the algorithm is in O(g4+gS(g)+K(g, g)+CK(g , g , g))and the space complexity of the result is in O(g2).

1. Apply Algorithm3.4.1to H0(X,L2(D1)) and H0(X,L2(D2)) toobtain H0(X,L4(D1 D2)).

2. Apply Algorithm 3.4.2 to the modules H0(X,L3), H0(X,L) andH0(X,L4(D1 D2))to obtain H0(X,L3(D1 D2)).

3. Choose a non-zero element s ofH0(X,L3(D1 D2)).

4. Apply Algorithm 3.4.1 to s and H0(X,L2) to obtain the moduleH0(X,L5(D1 D2 E)).

5. Apply Algorithm3.4.2to the modules H0(X,L2), H0(X,L3(D1 D2))and H0(X,L5(D1 D2 E))to obtain

H

0

(X,L

2

(E)) = (H

0

(X,L

5

(D1D2E)) :H

0

(X,L

3

(D1D2))).

6. Return H0(X,L2(E))and the section s.

Proof. Asdeg(L2(Di)) 2g+1fori = 1, 2, Algorithm3.4.1can be appliedin step1to obtainH0(X,L4(D1D2)). Asdeg(L3(D1 D2)) 2g + 1,Algorithm3.4.2in step2 calculates H0(X,L3(D1 D2)). The element schosen in step3corresponds toH0(X,L3(D1D2E)), which has degree3(2g + 1), and so Algorithm3.4.1givesH0(X,L5(D1 D2 E))in step4.Nowdeg(L5(D1 D2 E)) = 5(2g+ 1) and so Algorithm3.4.2in step5producesH0(X,L2(E)).

The space complexity of each submodule appearing in the algorithm isO(g2). Since each step of the algorithm makes at most one call to eitherAlgorithm3.4.1or3.4.2, the time complexity is in O(g4 + gS(g) + K(g, g) +CK(g , g , g)). As H0(X,L2(E)) is a submodule of H0(X,L2), its spacecomplexity is O(g2).

All of the familiar arithmetic operations on Pic0X(S)can be described interms of the Addflip operation given by Algorithm3.6.3; we show this inthe following algorithm.


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Algorithm 3.6.4(Negation, addition, subtraction, and equality).Letxandy be elements ofPic0(X)given as submodules ofH0(X,L2). Then we cancalculate the negation ofx, the sum ofx and y, the difference ofx and y,and whether x and y are equal. If the inputs have space complexities inO(g2), then each operation has time complexity in O(g4 + gS(g) + K(g, g) +CK(g , g , g)) and the results have space complexities in O(g2) (except theequality test which is simply true or false).

(i) Negation: Apply Algorithm3.6.3toxand the result of Algorithm3.6.1to produce x 0 = x.

(ii) Addition: Apply Algorithm3.6.3toxand y and then negate the result,giving x + y= (x y).

(iii) Subtraction: Negate xand apply Algorithm3.6.3toxand y, givingx y = (x) y.

(iv) Equality: Subtractyfromx, then apply Algorithm3.6.2to test whetherthe result is zero.

Proof. That the calculations are correct is clear. The dominant calculationin each case is Algorithm3.6.3, whose time complexity is in O(g4 + gS(g) +K(g, g) + CK(g , g , g))and whose results have space complexity in O(g2).

We can define scalar multiplication by an arbitrary integer using the addi-tion and negation defined in Algorithm3.6.4. Implemented as the well-knowndouble-and-add algorithm based on Horners rule for scalar multiplication inan additive group, its time complexity is in O(log(n)(g4 + gS(g) + K(g, g) +CK(g , g , g)))operations in R.


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Part II

Spaces of sections on algebraicsurfaces


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Chapter 4

Cohomology of surfaces

In this chapter we analyse the structure of the cohomology groups of divisorson several classes of surfaces, namelyC CandSym2(C)for a hyperellipticcurve C of genus at least two. We prove decompositions of the spaces ofglobal sections of divisors on these surfaces and deduce formul for theirdimensions.

Throughout this chapter and the next, bysurfacewe will mean a nonsin-gular projective algebraic surface over an algebraically closed field.

4.1 Intersection theory on surfacesIn this section we recall the basic facts of intersection theory on surfaces.The primary reference is Hartshorne [17,Chapter V].

LetCandDbe two prime divisors on a surfaceXand letxbe a point inC D. Letfandg be respectively local equations for CandD at x. We saythat Cand D intersect transversallyat xiff and g generate the maximalideal mx ofOX,x; this implies, in particular, that C and D are nonsingularat x.

Theorem 4.1.1. Let X be a surface and let C and D be divisors on X.There exists a unique bilinear symmetric pairing

Div(X) Div(X) Z,

denoted by(C, D) C D, such that the following conditions are satisfied:

(i) IfCandDmeet transversally, thenC D= #(C D), the number ofpoints of intersection ofCandD.


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4.1 Intersection theory on surfaces 45

(ii) For allC1 andC2 inDiv(X), ifC1 rat C2, thenC1 D =C2 D. In

particular, the pairing induces a well-defined bilinear symmetric pairing

Pic(X) Pic(X) Z

which we also denote by([C], [D]) C D.

Proof. Hartshorne[17,Theorem V.1.1].

We call C D the intersection numberof the divisors C and D. Theself-intersection numberofC isC Cwhich we will often write as C2 whenthere is no chance for confusion with the scheme product of curves. More

generally we write Cn

=C C C(ntimes).Proposition 4.1.2(Adjunction formula). LetXbe a surface and letKXbea canonical divisor onX. Then for any nonsingular curveCof genusg onX,

2g 2 =C (C+ KX).

Proof. See Hartshorne [17,Proposition V.1.5].

Proposition 4.1.3. Letf: X Ybe a surjective, projective morphism ofNoetherian integral schemes and suppose n = [K(X) : fK(Y)] is finite.

Then for any Cartier divisorDonY,ff

D= nD.

Proof. See Liu [26,Proposition 9.2.11].

Proposition 4.1.4.Letf: X Ybe a dominant map of smooth projectivesurfaces. Thenn= [K(X) :fK(Y)]is finite and for any divisorsCandDonY,

fC fD= nC D.

Proof. See Liu [26,Proposition 9.2.12(c)].

Proposition 4.1.5(Riemann-Roch for surfaces). LetXbe a surface, letDbe a divisor onX, and letKXbe a canonical divisor onX. Then

(D) =1

2D (D KX) + (OX).

Proof. See Hartshorne [17,Theorem V.1.6].


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4.2 Equivalence classes of divisors 46

4.2 Equivalence classes of divisors

In this section we introduce the notions of algebraic and numerical equiva-lence, and the corresponding quotient groups: the Nron-Severi group andthe numerical equivalence group of a variety. We will analyse the structureof these groups in the case of squares and symmetric squares of hyperellipticcurves. Many of the results in the first part of this section are true in muchgreater generality; see Kleiman[24, Sections 46] for details.

Let Xbe a nonsingular projectively variety. Recall from Section1.4thatthe Picard group ofXis defined to be the group of isomorphism classes of in-vertible sheaves onX. By Theorem1.4.3,Pic(X)is isomorphic to CaCl(X),

the group of equivalence classes of divisors onXmodulo rational equivalence.In the interests of brevity, we will (re)define Pic(X)to bePic(X) = CaCl(X),and ifDis a divisor on X, we will write [D]for the class in Pic(X).

Let Xbe a surface and let D be a divisor on X. Then D is said to bepre-algebraically equivalent to zero if there exists a nonsingular curve T, aneffective divisor E on X Twhich is flat over T, and two closed points sand t on T, such that

D= E (X {s}) E (X {t}).

Two divisors are said to be pre-algebraically equivalent if their difference

is pre-algebraically equivalent to zero. Two divisors D and D on X aresaid to be algebraically equivalent if there exists a finite sequence D0 =D, D1, . . . , Dn =D

of divisors with Di pre-algebraically equivalent to Di+1for each i = 0, . . . , n 1. IfDand D are algebraically equivalent we writeD algD.

Lemma 4.2.1. LetD1 andD2 be divisors on a surfaceX. IfD1 rat D2,thenD1alg D2.

Proof. Let D = div(f)be a principal divisor on Xand let D = div(uf v)be a divisor on X P1 were uand v are the coordinates on P1. Then D is

flat over P1 and

D (X {(1 : 0)}) D (X {(0 : 1)}) = div(f) div(1) =D

as required

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Algorithmic Aspects of Hyperelliptic Curves