Some results for rank 3 vector bundles, reflexive sheaves and instanton bundles in the complex projective space of dimension 3 : Alguns resultados sobre fibrados, feixes reflexivos e fibrados instanton de posto 3 no espaço projetivo complexo de dimensão 3

CAMPINAS

Instituto de Matemática, Estatística e

Computação Científica

LUIZ CARLOS DA SILVA SOBRAL

Some results for rank 3 vector bundles,

reflexive sheaves and instanton bundles in the

complex projective space of dimension 3

Alguns resultados sobre fibrados, feixes

reflexivos e fibrados instanton de posto 3 no

espaço projetivo complexo de dimensão 3

Campinas

2019

Some results for rank 3 vector bundles, reflexive sheaves

and instanton bundles in the complex projective space of

dimension 3

Alguns resultados sobre fibrados, feixes reflexivos e

fibrados instanton de posto 3 no espaço projetivo

complexo de dimensão 3

Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Matemática.

Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computing of the University of Campinas in partial ful-fillment of the requirements for the degree of Doctor in Mathematics.

Orientador: Marcos Benevenuto Jardim

Coorientador: Simone Marchesi

Este exemplar corresponde à versão

fi-nal da Tese defendida pelo aluno Luiz

Carlos da Silva Sobral e orientada pelo

Prof. Dr. Marcos Benevenuto Jardim.

Campinas

2019

Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467

Sobral, Luiz Carlos da Silva,

So12s SobSome results for rank 3 vector bundles, reflexive sheaves and instanton bundles in the complex projective space of dimension 3 / Luiz Carlos da Silva Sobral. – Campinas, SP : [s.n.], 2019.

SobOrientador: Marcos Benevenuto Jardim. SobCoorientador: Simone Marchesi.

SobTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.

Sob1. Fibrados vetoriais. 2. Instantons. 3. Teoria dos feixes. 4. Teoria de módulos. 5. Geometria algébrica. I. Jardim, Marcos Benevenuto, 1973-. II. Marchesi, Simone, 1984-. III. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. IV. Título.

Informações para Biblioteca Digital

Título em outro idioma: Alguns resultados sobre fibrados, feixes reflexivos e fibrados

instanton de posto 3 no espaço projetivo complexo de dimensão 3

Palavras-chave em inglês: Vector bundles Instantons Sheaf theory Moduli theory Algebraic geometry

Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora:

Marcos Benevenuto Jardim [Orientador] Aline Vilela Andrade

Renato Vidal da Silva Martins Valeriano Lanza

Danilo Dias da Silva

Data de defesa: 03-05-2019

Programa de Pós-Graduação: Matemática

Identificação e informações acadêmicas do(a) aluno(a)

- ORCID do autor: https://orcid.org/0000-0003-3923-1941 - Currículo Lattes do autor: http://lattes.cnpq.br/3827182830559979

pela banca examinadora composta pelos Profs. Drs.

Prof(a). Dr(a). MARCOS BENEVENUTO JARDIM

Prof(a). Dr(a). DANILO DIAS DA SILVA

Prof(a). Dr(a). RENATO VIDAL DA SILVA MARTINS

Prof(a). Dr(a). ALINE VILELA ANDRADE

Prof(a). Dr(a). VALERIANO LANZA

A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica.

Esta tese foi um fruto de um trabalho duro e árduo. Passei por várias dificuldades no decorrer do tempo em que estive em Campinas. Porém eu tive várias pessoas que me ajudaram e me deram forças para suportar esse peso, no qual sem eles não chegaria a lugar algum.

A primeira pessoa que devo agradecer desde o princípio é de fato Deus, que desde o inicio me ajudou encaminhando todos as pessoas que foram importantes e que de alguma forma me ajudaram nesse processo.

Segundo agradeço aos meus pais por me criarem e me ensinaram as coisas da vida e estiveram sempre presentes na minha vida.

A terceira pessoa que eu devo agradecer é obviamente a Marcos Jardim, pois além de ser um ótimo orientador, ele foi uma pessoa incrivelmente paciente. Em outras palavras, ele além de ser uma pessoa muito bondosa e tolerante ele é uma matemático no qual as gerações futuras devem se espelhar, principalmente nós brasileiros que trabalhamos na área de Geometria Algébrica.

Além dessas pessoas, eu agradeço a todos os meus amigos, Aelton Feitosa, Anselmo Feitosa, Cícero, Raoni que me acompanharam durante minha infância. Aos amigos do ensino médio: Adeval, Daniel Santana, Daniel Valadão, Alysson Ancelmo, o seu primo Douglas, Vieira, Lucas, Renan Souza. Aos professores Washington, Leopoldo, Anselmo, Natanael, Valdemberg, Kalasas Vasconcelos, Danilo Felizardo, Evilson e Paulo Rabelo, Simone Marchesi, Fabricio Matino, Alcíbiades Rigas, Luiz San Martin e Francesco Mercuri.

Não podemos nos esquecer dos grandes colegas de batalha que me acompanha-ram a minha graduação, mestrado e doutorado, como Reinan Ribeiro, Patricia, Paulo Eduardo, Moisés, Nyorison, Diego, Jonison e Ginaldo. Do mestrado são Charlene, Jus-sineide, Jeocastria, Samuel. Do doutorado são Aline, Charles Almeida, Charles Martins, Thiago Estevez, Samuel, Miquéias, Fabio Rodrigues, Danilo , Claudemir Fideles, José e tantos outros amigos.

Um agradecimento especial que devo compartilhar nesse texto é para meu amigo Marcelo, pois além de conhecê-lo como pessoa ele foi a única pessoa que realmente compartilhei minha maneira de pensar matemática e me compreendeu de uma certa forma. Um grande agradecimento a ele por ser a pessoa batalhadora como ele é. Acredito que ele será um grande matemático. Também não posso esquecer de um grande amigo meu Marcos Azevedo, trocamos muitas idéas, pena que o doutorado não deu certo para ele,

abraço para ele.

O presente trabalho foi realizado com apoio da Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Código de Financiamento 001 e com o apoio da bolsa CNPQ com o processo 140727/2016-7.

O objetivo principal dessa tese é a caracterização dos fibrados instanton de posto três no espaço projetivo complexo de dimensão três sem seções globais. O método usado para fazer essa caracterização é a correspondência de Hartshorne-Serre relacionando esses fibrados instanton a curvas racionais, como também calculamos a dimensão do espaco de moduli desses fibrados, no final damos um exemplo de um ponto suave desse espaco de moduli.

The main goal of this thesis is the caracterization of rank 3 nstanton bundles in the complex projective space of dimension 3 without global sections. The used tool is the Hartshorne-Serre correspondence connecting these instanton bundles to the rational curves. Also we compute the dimension of moduli space of these instanton bundles and finally we found a example of a nonsingular point of this moduli space

Introduction . . . 10

1 BASIC TOPICS . . . 11

1.1 Hartshorne-Serre correspondence . . . 11

1.2 Reflexive sheaves . . . 14

1.3 Stability of reflexive sheaves . . . 15

1.4 Spectra of reflexive sheaves. . . 17

1.5 Introduction to linear sheaves and instanton sheaves . . . 20

2 KNOWN RESULTS ABOUT RANK 2 VECTOR BUNDLES . . . . 22

2.1 Splitting criterion . . . 22

2.2 Stability criterion . . . 24

2.3 Chern classes of a rank 2 vector bundle and degree and genus of its associated curve . . . 24

3 ANALOGOUS RESULTS FOR RANK 3 VECTOR BUNDLES . . . 27

3.1 Stability criterion . . . 27

3.2 Chern classes of rank 3 locally free sheaf and degree and genus of its associated curve . . . 29

3.3 Relation between rank 2 and rank 3 vector bundles associated a subcanonical curve . . . 30

3.4 Splitting criterion . . . 33

3.5 Expected dimension of moduli space of rank 3 vector bundles with Chern classes c1, c2, c3 . . . 36

4 THE MODULI SPACE OF RANK 3 INSTANTON BUNDLES WITHOUT GLOBAL SECTIONS OF CHARGE 2 . . . 38

4.1 Rank3 instanton sheaves without global sections and rank 2 stable reflexive n sheaves . . . 38

4.2 Instanton sheaves of charge n without global sections . . . 46

4.3 Important example for this thesis . . . 48

REFERÊNCIAS . . . 56

Introduction

The main goal of this thesis is to study rank 3 instanton vector bundles without global sections of charges 2 and 3 on P3. The fundamental tools used in this thesis are the concept of spectra of semistable reflexive sheaves and the Hartshorne-Serre correspondence. Classifying instanton bundles of arbitrary charge is a interesting problem in algebraic geometry and to find examples of stable bundles, indecomposable bundles and generalizations of the Hartshorne-Serre correspondence are surely examples of hard problems (See for instance (ARRONDO,2007) and (HARTSHORNE,1979)). There exists a vast literature about rank 2 vector bundles on Pn but the literature about rank 3 vector bundles or more general vector bundles of arbitrary rank is not very large. On the other hand, the book used as the main reference in this thesis by Schneider, Spindler and Okonek (See (OKONEK et al., 1980)) is of great importance in the study of vector bundles of

arbitrary rank in general projective space.

Recently, Marcos Jardim and Michael Gargate characterized the indecomposable rank 3 instanton vector bundles of charge 1 (See (JARDIM, 2005)). This thesis is a continuation of this article in the sense that it tries to classify rank 3 instanton bundles of small charge different of 1 by relating the instanton bundles without global sections of charges 2 and charges 3 with rational curves in P3 of degrees 5 and 6 respectively by considering the Hartshorne-Serre correspondence.

Moreover, we set a correspondence between stable linear reflexive sheaves of rank 2 and Chern classes (1, 3, 3) and Chern Classes (1, 4, 4) and rank 3 instanton bundles without global sections of charges 2 and charge 3 respectively. It is interesting to notice that the results obtained in this thesis for the case of charge 3 were not as complete as for the case of charge 2 since the moduli space of reflexive linear sheaves with Chern classes (1, 4, 4) is not as well known as the moduli space of Chern classes (1, 3, 3). For instance, in case of charge 2 we found the dimension of moduli space of rank 3 instanton bundles without global sections and an example of a nonsingular point in this moduli space by using known results about the dimension of the moduli space of stable linear reflexive sheaves of Chern classes (1, 3, 3) while the same was not possible for charge 3. Finally, we mention the crucial work of Mei-Chu-Chang (See for instance (CHANG, 1984)) which helped us to find the above results through the mentioned correspondence.

1 Basic topics

1.1

Hartshorne-Serre correspondence

In this section we present some definitions and properties regarding the Hartshorne-Serre correspondence which consists on associating subschemes Y of codimen-sion 2 in a nonsingular algebraic variety X to pairs (E, L), where E is a vector bundle and

L is a line bundle in X. In Hartshorne’s paper ( See (HARTSHORNE,1980)) it is proved that this correspondence works in case X = P3 and Y a curve in X while the general case can be found in Arrondo’s paper( See (ARRONDO, 2007)). One of the features of the Hartshorne-Serre correspondence is the possibility to express a rank r vector bundle E on

X by the following exact sequence

0 → (r − 1)OX → E → IY(k) → 0

where IY is the ideal sheaf of a codimension 2 subscheme Y in X and k is the first Chern

class of E. In other words, it consists in to get some information of E in terms of Y and this can be considered as the heart of Hartshorne-Serre Correspondence. In particular, it can be used in the case in which X = P3, E is a rank 3 vector bundle in X and Y is a curve in X. Let us begin with the following definition.

Definition 1.1. Let X be a projective variety over C. Consider a codimension r subscheme

Y in X and IY /X its ideal sheaf. If there exists a covering {Uj}j∈J of Y such that IY /X(Uj) is generated by r elements, then Y is called a locally complete intersection subscheme. If IY /X(Y ) is generated by r elements then Y is said to be a globally complete intersection subscheme.

This definition is a crucial hypothesis in the Hartshorne-Serre Correspondence. Besides the concept of a globally complete intersection subscheme is also important because its associated vector bundle splits in the case of rank 2 vector bundles. In this thesis we prove that this result is not true for rank 3 vector bundles in general, but with a few additional hypothesis something can be done. Unfortunately in this thesis, we give only a sufficient condition for E to be split.

The next definition is important for the construction of the zero loci of a vector bundle since this zero loci is a determinantal variety of a morphism of vector bundles constructed by global sections of it. Recall that if E is a vector bundle then its associated curve Y is the zero loci of a section of E. This shows the importance of determinantal varieties and zero loci of vector bundles.

Definition 1.2. Consider the set M (m × n) of n × m matrices. Let 0 ≤ q ≤ min{m, n}.

Then the r−th determinantal Variety is the subset of m × n matrices of rank lower or equal than q.

Definition 1.3. Let E and F be rank m and n (respectively) vector bundles, φ : E → F be

a morphism and U be a trivializing open set. Then we get a morphism f : U → M (n × m) induced by φ. If Mk is the k-th determinantal variety ( 0 ≤ k ≤ min{m, n}) and Uk its inverse image then by gluing each Uk we get a variety Σ called the k − th determinantal variety associated to φ.

Definition 1.4. Let E be a rank r vector bundle such that H0(X : E) 6= 0 and let

φ1, · · · , φr−1 be the global sections of E. Then we can construct a morphism:

φ : (r − 1)OX → E

so the dependency loci of E is the (r − 2) − th determinantal variety of φ.

The following theorem is due to Hartshorne; it is the original Hartshorne-Serre correspondence considering the case of a rank 2 vector bundle on P3.

Theorem 1.5. Let Y be a curve in P3. Then Y is the dependency loci of a section σ of a vector bundle E if and only Y is a locally complete intersection subscheme and ωY is isomorphic to the restriction to Y of an invertible sheaf on P3. More precisely, for any fixed invertible sheaf L on P3, there exists a bijective correspondence between:

(i) The set of triples (E, s, φ) with a equivalence relation ∼, where E is a rank 2 vector bundle in P3, s is a global section of E whose dependency loci is codimension 2 variety in P3 and φ :

2

^

E → L is a isomorphism. The equivalence relation is

(E, s, φ) ∼ (E0, s0, φ0) if and only exists a isomorphism ψ : E → E0 and a element λ ∈ k − {0} such that s0 = λψ(s) and φ0 = λ2◦ (

2

^

ψ)−1

(ii) Pairs (Y, ξ) consisting of a local complete intersection curve Y in P3 and ξ : L ⊗ ω_P3⊗ OY → ωY is a isomorphism.

Proof: See (HARTSHORNE, 1978) Theorem 1.1 pg 232

Later Vogelaar generalized this construction for locally free sheaves of arbitrary rank (See (VOGELAAR,1978)) and yet another generalization for rank 2 reflexive sheaves on P3 was given by Hartshorne himself (See (HARTSHORNE, 1980)). The following theorem was proved by Arrondo by using elementary techniques and it is the main reference used today for the generalized Hartshorne-Serre correspondence (See (ARRONDO,2007)).

Theorem 1.6. Let X be a smooth algebraic variety and Y be a codimension 2 locally

complete intersection subscheme. Let N be the normal bundle of Y in X and L be a line bundle in X such that H2(X, L∗) = 0. We assume that

2

^

N ⊗ L∗|Y is generated by r − 1 global sections σ1, · · · , σr−1. Then there exists a rank r vector bundle E on X such that:

(i) r

^

E = L

(ii) E has r − 1 global sections t1, · · · , tr−1 whose dependency locus is Y and:

σ1t1+ · · · + σr−1tr−1 = 0

If H1(X, L∗) = 0 then E is unique up to an isomorphism.

Proof: See (ARRONDO, 2007) Theorem 1.1 pg 426.

The next lemma which shall be used a few times in this thesis requires the definition of the dualizing sheaf of a subscheme.

Definition 1.7. Let Y a subscheme of Pn of codimension r. We define the dualizing sheaf of Y by wY = E xtr(OY, ωPn), where ωPn = OPn(−n − 1).

Lemma 1.8. Let Y a codimension 2 closed subscheme of Pn. Then we have

Ext1_(I

Y(k), O) ∼= ωY(n − k + 1)

Proof: Consider the exact sequence:

0 → IY → O → OY → 0

Twisting by O(k) we have:

0 → IY(k) → O(k) → OY(k) → 0

Then, using this above exact sequence we have: Ext1_(O

Y(k), O) → Ext1(O(k), O) → Ext1(I(k), O) → Ext2(OY(k), O) →

Ext2(O(k), O)

Hence we conclude that Ext1_(I Y(k), O) ∼= E xt2(OY(k), O) ∼ = E xt2(OY(k), ωPn(n + 1) ∼ = O(−k) ⊗ E xt2(OY, ωPn) ⊗ OPn(n + 1) = ωY(n − k + 1) (1.1)

1.2

Reflexive sheaves

In this section we deal with reflexive sheaves. A coherent sheaf F is reflexive if the natural morphism F → F∗∗ is an isomorphism of sheaves. Reflexive sheaves are a generalization of vector bundles and as such a natural issue is whether we can adapt results regarding vector bundles to reflexive sheaves. An example is the Hartshorne-Serre correspondence between rank 2 reflexive sheaves and curves in P3 with some properties proved by Hartshorne. In this section, we present some results regarding reflexive sheaves used in this thesis.

Lemma 1.9. Let F be a rank 2 reflexive sheaf in Pn and c1 be its first Chern class. Then F∗ = F (−c1).

Proof: See (HARTSHORNE, 1980) proposition 1.10 pg 129

Lemma 1.10. Let F be a rank 2 reflexive sheaf in P3. Then c3 = E xt1(F, O). In particular, c3 ≥ 0 and F is locally free, if and only if c3 = 0.

Proof: See (HARTSHORNE, 1980) Theorem 2.6 pg 131.

Theorem 1.11. Fix a integer c1. Then there is a one-to-one correspondence between: (i) Pairs (F , s), where F is a rank 2 reflexive sheaf in P3 with first chern class c1 and s

is a global section of F , whose dependency loci has codimension 2, and

(ii) Pairs (Y, ξ), where Y is a Cohen-Macaulay curve in P3, generically complete inter-section, and ξ is a global section in ωY(4 − c1) such that generates ωY(4 − c1) except at finitely many points.

Futhermore, under this correspondence we have c2 = d and c3 = 2g − 2 + d(4 − c1), where d and g are respectively the degree and genus of a curve Y .

Proposition 1.12. Let F be a rank 2 reflexive sheaf in P3 with Chern classes c1, c2 and c3, then 3 X i=0 (−1)idim Exti(F , F ) = 2c12− 8c2+ 4 .

Proof: See (HARTSHORNE, 1980) Proposition 3.4 pag 135.

This last proposition is interesting because it gives information on the dimension of the tangent space of a moduli space of stable reflexive sheaves. It will be used in Chapter

3.

1.3

Stability of reflexive sheaves

In this section we explain a very important concept in algebraic geometry, the Mumford-Takemoto stability for coherent sheaves. It will be use in latter chapters in this thesis (mainly in the final chapter).

Definition 1.13. Let F be a torsion free sheaf in Pn and c1 be its first Chern class. We define the slope of F as the quotient µ(F ) = c1

rank(F ).

Definition 1.14. A torsion free in Pn is a stable (semistable) sheaf if for all torsion free subsheaf L with rank lower than F , we have µ(F ) > µ(L)(resp µ(F ) ≥ µ(L)).

Theorem 1.15. Let E be a torsion free sheaf in Pn and m ∈ Z. Then E is stable (semistable) if and only E(m) is stable (resp. semistable).

Proof: See (OKONEK et al., 1980) pg 83 Lemma 1.2.4.

Theorem 1.16. Let E be a reflexive sheaf in Pn. Then E is stable(semistable) if and only E∗ is.

Proof: See (OKONEK et al., 1980) Lemma 1.2.4 pg 83.

Definition 1.17. Let E be a rank r vector bundle in Pn and c1 be its first Chern class. Then there exists k ∈ Z such that c1(E(k)) ∈ {0. − 1, · · · , −r + 1}. In this case we say that E(k) is the normalized vector bundle of E.

The following criterion of stability is useful for rank 2 locally free sheaves.

Theorem 1.18. Let F be a rank 2 reflexive sheaf, which is normalized so that (i.e c1 = 0 or −1). Then F is stable if and only H0(F ) = 0. If c1 = 0, then F is semistable if, and only if H0(F (−1)) = 0

Proof: See (HARTSHORNE, 1980) Lemma 3.1 pg 133.

There is an analogous criterion of stability (semistablity) for rank 3 reflexive sheaves.

Theorem 1.19. Let E be a normalized rank 3 locally free sheaf in Pn. Then E is stable if and only if

(i) H0_(Pn, E) = H0_(Pn, E∗) = 0, if 3 divides c1(E).

(ii) H0_(Pn, E) = H0_(Pn, E∗(−1)) = 0, if 3 does not divide c1(E).

E is semistable if and only if H0_(Pn, E(−1)) = H0_(Pn, E∗(−1)) = 0. Proof: See (OKONEK et al., 1980) Remark 1.2.6 pg 85.

Theorem 1.20. If F is a rank 2 normalized stable reflexive sheaf. Then

dim Ext1(F , F ) − dim Ext2(F , F ) =  



8c2− 3, se c1 = 0,

8c2− 5, se c1 = −1

Proof: See (HARTSHORNE, 1980) Remark 3.41 pg 136.

Now we present a first example of a stable locally free sheaf obtained from the Hartshorne-Serre correspondence. This example is important because it shows the main idea of this thesis, namely the connection between rank 2 and rank 3 vector bundles associated to a local complete intersection curve.

Good Example 1.21. Let Y = L1∪ L2 be a disjoint union of two lines in P3. Each line Li, i = 1, 2 is Li = Hi ∩ Hi0 as intersection of two hyperplanes in P

3_{. The normal sheaf} of Y in P3 is N_{Y /P}3|_L

i = NHi/P3 ⊕ NHi0/P3 = OLi(1) ⊕ OLi(1), hence detNY /P3|Li = OLi(2).

Taking L = O(2) we get:

0 → O → E → IY(2) → 0

the Chern classes are c1 = c2 = 2, then c1(E(−1)) = 0 and c2(E(−1)) = 1, in addition E is stable. Therefore E(−1) ∼= N where N is a null- correlation bundle. Hence the rank 2

vector bundle associated to a disjoint union of two lines is a null-correlation bundle twisted by O(1).

Now let us approach some ideas about null correlation bundles. Null correlation bundles have many properties as:

c1(N ) = 0 c2(N ) = 1 N∗ = N H1(N (k)) =    0, se k 6= −1, 1, se k = −1

And H0(N (k)) = 0 if and only if k ≤ 0. Then Ext1(N, O(−1)) = H1(N∗(−1)) ∼=

H1(N (−1)). Whence exists a rank 3 vector bundle F such that:

0 → O(−1) → F → N → 0

This sequence gives rise to the following commutative diagram:

0  0  0  0 _// O  0 //O //F (1)  t _// N (1)  //₀ F (1)  u _// IL1∪L2(2)  0 0

By using the snake lemma we get:

O → 2O → F (1) → IL1∪L2(2) → 0

Hence we have a first example of a relation between rank 2 and rank 3 vector bundles associated to a curve Y . This kind of relation shall be more explored in the sequel.

1.4

Spectra of reflexive sheaves

The concept of spectra of reflexive sheaves was used in the thesis to calculate the cohomologies of a sheaf on P3. In general we can define spectra for a torsion free sheaf on P3 but in this section we explain some aspects regarding the spectra of rank 2 and rank 3 reflexives sheaves.

Theorem 1.22. Consider a rank 2 normalized reflexive sheaf F . If H0(F (−1)) = 0, then

there exists an unique set of integers {k1, . . . , kc2}, with the following properties (where H

denotes the sheaf MO_P1(k_i)).

(i) h1_(P3, F (l)) = h0_(P1, H(l + 1)),for l ≤ −1

(ii) h1_(P3, F (l)) = h0_(P1, H(l + 1)), for l ≥ −3 if c1 = 0 and l ≥ −2 if c1 = −1

The set {k1, . . . , kc2} is called the spectra of F and is denoted by kF.

Proof: See (HARTSHORNE, 1980) Theorem 7.1 pg 151.

Theorem 1.23. With the hypotheses of Theorem1.22, assume furthermore F locally free, then {−ki} = {ki} if c1 = 0 and {−ki} = {ki + 1} if c1 = −1.

Proof: See (HARTSHORNE, 1980) Proposition 7.2 pg 152. Theorem 1.24. With the hypotheses of Theorem 1.22. Then

(i) c3 = −2 c2 X i=1 ki, if c1 = 0 (ii) c3+ c2 = −2 c2 X i=1 ki, if c1 = −1

Proof: See (HARTSHORNE, 1980) Theorem 7.3 pg 153.

Theorem 1.25. Let F be as Theorem 1.22, and {−ki} by the its spectra. a) Assume that H0(F (−1)) = 0

1) If there exists a k > 0 in the spectra, then 1, 2, . . . k also occur in the spectra 2) If there exists a k < 0 in the spectra, then −1, −2, . . . k also occur in the spectra

if c1 = 0 and −2, . . . , k if c1 = −1. b) Assume F is stable

1) If there exists a k > 0 in the spectra, then 0, 1, 2, . . . k also occur in the spectra 2) If there exists a k < 0 in the spectra, then −1, −2, . . . k also occur in the spectra if c1 = 0 and −1, −2, . . . , k if c1 = −1. Furthermore, if c1 = 0 then either 0 occurs, or −1 occur at least twice.

Proof : See (HARTSHORNE, 1980) Theorem 7.5 pg 155,

Theorem 1.26. Let F be a rank 3 semistable sheaf. We define spectra of F as the sequence (k1, . . . , km) such that(where H denotes the sheaf

M

O(ki)). (i) h1(P3, F (l)) = h0_(P1, H(l + 1)), for l ≤ −1

(ii) h1_(P3, F (l)) = h0_(P1, H(l + 1)), for l ≥ −3 if c1 = 0 and l ≥ −2 if c1 = −1, −2

Proof: See (COANDA, 1986) pg 165,

Theorem 1.27. With the hypotheses of Theorem 1.26 . Then: (i) m = c2, if c1 = 0 and m = c2− 1, if c1 = −1 or c1 = −2. (ii) c3 = −2 m X i=1 ki , if c1 = 0 (iii) c3+ c2 = −2 m X i=1 ki, if c1 = −1 (iv) c3+ 2c2− 2 = −2 m X i=1 ki, if c1 = −2

Proof: See (COANDA, 1986) pg 165,

Theorem 1.28. Let F be a semistable rank 3 reflexive sheaf and (k1, . . . , km) its spectra.

(i) Suppose c1 = 0. If there exists a i such that ki ≤ −1(resp ki ≥ 1). Then ki, ki +

1, . . . , −1 (resp 1, 2, . . . , ki) occur in the spectra.

(ii) Suppose c1 = −1 or −2. If there exists a i such that ki ≤ −1(resp ki ≥ 0). Then k1, ki+ 1, . . . , −1 (resp 1, 2, . . . , ki) occur in the spectra.

Proof: See (COANDA, 1986) Proposition 2.1 pg 166.

Example 1.29. Let E be a rank 2 stable reflexive sheaf with Chern classes (−1, 4, 4). Let

kE = (k1, . . . , km) its spectra. Then by the Theorem 1.27, we have

4 + 4 = −2 4 X i=1 ki then: k1 + k2+ k3+ k4 = −4

then k1 ≤ −1 and k4 ≥ −1. Theorem 1.25 b) implies that k1 ≥ −4, again Theorem 1.25 implies that kE = (−2. − 1, −1, 0) or kE = (−1. − 1, −1, −1), because the possibilities k1 = −4 and k1 = −3 cannot occur.

Example 1.30. Let E be a rank 3 stable sheaf with Chern classes (−2, 3, 4). then by the Theorem 1.27 we have: −4 + 2 × 3 − 2 = −2 3 X i=1 ki

Hence k1+k2+k3 = 0, then k1 ≤ 0 and k3 ≥ 0, by the Theorem1.28we conclude that k1 ≥ −3, but again by the Theorem 1.28 and the above equation, we conclude that the possibilities k1 = −2 or k1 = −3 cannot occur, then or kE = (−1, 0, 1) or kE = (0, 0, 0).

1.5

Introduction to linear sheaves and instanton sheaves

Monads are object of the great importance in algebraic geometry since due to a result by Horrocks every rank 2 vector bundle is the cohomology of a monad whose elements are sums of line bundles ( see (HORROCKS, 1964)).

Definition 1.31. A monad in Pn is the following complex of locally free sheaves

0 //A α //B β //C //0

such that β ◦ α = 0. The sheaf E = ker(β)

im(α) is called the cohomology of monad.

Definition 1.32. A monad of the form:

0 //uO_Pn(−1) α //wO

Pn β _//

uO_Pn(1) //0

is called a linear monad. The cohomology of a linear monad is called a linear sheaf.

Now we present the definition of instanton.

Definition 1.33. An instanton sheaf E on P3 is a sheaf satisfying the following cohomology conditions

H0(E(−1)) = H1(E(−2)) = H2(E(−2)) = H3(E(−3)) = 0

and such that its first Chern class is 0. The second Chern class is called the charge of E.

In this section the main idea is to relate instanton sheaves and linear sheaves which is accomplished by the following results:

Theorem 1.34. E is a linear sheaf in P3 if and only:

Proof: See (JARDIM, 2005) Theorem 3 pg 5 and Proposition 2 pg 4.

Theorem 1.35. E is an instanton sheaf if and only E is the cohomology of a monad of

the form

0 //nO_Pn(−1) α //(r + 2n)O

Pn β _//

nO_Pn(1) //0

2 Known results about rank 2 Vector bundles

2.1

Splitting criterion

In this chapter we present some results about rank 2 locally free sheaves on P3. These results are known ( See (HARTSHORNE,1978) corollary 1.2 pg 233 and Proposition 2.1 pg 236) but we prove them here for convenience of the reader.

Theorem 2.1. Let E be a rank 2 vector bundle on P3, σ be a global section of E and Y its dependency loci. Then E splits as a sum of line bundles if and only if Y is a global complete intersection.

Proof: If E splits, then we can suppose E = O(D1) ⊕ O(D2), where D1 and D2 are

divisors in X. So each global section of E is a pair (σ1, σ2), where σi is a section of O(Di), i = 1, 2. Then Y = {σ = 0} = {σ1 = σ2 = 0} = D1∩ D2.

Reciprocally ,we assume Y = D1∩ D2 where D1 and D2 are divisors in X. The

vector bundle E = O(D1)⊕O(D2) has a vanishing section in Y and L ∼= 2

^

E = O(D1+D2).

The associated exact sequence is

0 → O(−D1− D2) → O(−D1) ⊕ O(−D2) → IY /X → 0

Twisting this sequence by L we get

0 → O → O(D1) ⊕ O(D2) → IY /X⊗ L → 0

and dualizing the above sequence we get

0 → O(−D1− D2) → O(−D1) ⊕ O(−D2) → O → E xt1(IY /X⊗ L, O) → 0

Since the last morphism in the exact sequence is surjective, we have

Ext1_(I

Y /X⊗ L, O) ∼= OY

consequently H0(E xt1(IY /X ⊗ L, O)) = H0(OY) ∼= k,because Y is a global complete

intersection curve. Hence the element ξ ∈ Hom(L ⊗ ω_P3⊗ O_Y, ω_Y) is unique up to a scalar

multiple. then

.

Now consider a rank 3 locally free sheaf on P3 and its associated curve Y in P3. A natural question is to wonder if E splits then Y is a global local intersection. And if the answer is affirmative does the reciprocal hold?

Example 2.2. Let X = P3. Consider the exact sequence

0 //2O A //3O(1) //IY(3) //0 whose first morphism is given by the matrix

A =      x y z w y z      associated variety is Z(xw − yz, z2 _{− yw, y}2_{− xz)}

this variety is a twisted cubic, but a twisted cubic is not a global complete intersection.

Example 2.3. Consider the Euler exact sequence

0 → O → 4O(1) → T_P3 → 0

By using this exact sequence we get a commutative diagram

0  0 //0 0  0 _// 2O(1) i  id _// 2O(1) b◦i  //₀ 0 //O a //4O(1)  b _// T_P3  //₀ 2O(1)  Coker(η) 0

Moreover by the snake lemma we get

obtaining

0 → 2O(1) → T_P3 → I_L(2) → 0

We get that the line L is a global complete intersection but T_P3 does not split.

We then conclude that the global complete intersection condition is neither necessary nor a sufficient condition for the decomposition of the associated rank 3 associated locally free sheaf. But in the next chapter we find a sufficient condition that guarantees the decomposition of the associated rank 3 locally free sheaf.

2.2

Stability criterion

The following theorem is a simple stability theorem for rank 2 locally free sheaves on P3.

Theorem 2.4. Let Y be a curve in P3 and E be the associated rank 2 vector bundle. Then E is stable if and only if c1(E) > 0 and Y is not contained in a surface of degree lower than or equal to c1(E)

2 .

Proof: See (OKONEK et al., 1980) Lemma 1.3.4. pg 93.

2.3

Chern classes of a rank 2 vector bundle and degree and genus

of its associated curve

In this section we prove a result connecting the degree and genus of a curve Y in P3 to the Chern classes of the associated vector bundle.

Theorem 2.5. Let Y be a locally complete intersection curve in P3, E be its associated rank 2 vector bundle on P3 and c1 and c2 be the Chern classes of the bundle. Then d = c2 and

c2(c1− 4) = 2g − 2

where d and g are respectively the degree and genus of a curve Y .

Proof: it is known that

ch(E) = 2 + c1H + 1 2(c1 2_{− 2c} 2)H2+ 1 6(c1 3_{− 3c} 1c2)H3

where H4 _{= 0 the equivalence class of a hyperplane in P}3 _{in Q[H]/(H}4). Since ch(O(m)) = 1 + mH + m 2 2 H 2₊m3 6 H 3 we obtain ch(E(m)) = = ch(E)ch(O(m)) = 2 + c1H + 1 2(c1 2_{− 2c} 2)H + 1 6(c1 3_{− 3c} 1c2)H3) × (1 + mH + m2 2 H 2₊ m3 6 H 3₎ = 2+(c1+2m)H+( 1 2(c 2 1−2c2)+mc1+m2)H2+( 1 6(c1 3_−3c 1c2)+ m 2(c1 2_−2c 2)+ m2_c 1 2 + m3 3 )H 3

using Riemann-Roch formula, we get

χ(E(m)) = (ch(E(m))td(P3))[H3_]= = 1 6(c1 3_{− 3c} 1c2) + m 2(c1 2_{− 2c} 2) + m2c1 2 + m3 3 + (c1 2_{− 2c} 2) + 2mc1+ 2m2+ 11c1 6 + 11m 3 + 2

Now considering the exact sequence

0 → O → E → IY(c1) → 0

we obtain

χ(E(m)) = PE(m) = PO(m) + PIY(m + c1)

where PE is the Hilbert polynomial of E. Using the exact sequence

0 → IY → O → OY → 0

we conclude

PIY(m + c1) = PO(m + c1) − POY(m + c1)

. hence

χ(E(m)) = PO(m) + PO(m + c1) − POY(m + c1) . By pO(m) = m + 3 3 ! and POY = md + 1 − g.

we can conclude that

χ(E(m)) = m + 3 3 ! + m + c1+ 3 3 ! − (m + c1)d − 1 + g .

Comparing the coefficients in m this formula with

χ(E(m)) = 1 6(c1 3_−3c 1c2)+ m 2(c1 2_−2c 2)+ m2_c 1 2 + m3 3 +(c1 2_−2c 2)+2mc1+2m2+ 11c1 6 + 11m 3 +2 we get the result.

3 Analogous results for rank 3 vector bundles

3.1

Stability criterion

Given a curve Y ∈ P3 which satisfies the hypothesis of the Hartshorne-Serre Correspondence we consider in this chapter the rank 3 locally free sheaf E associated to Y by the result of Arrondo (see (ARRONDO, 2007)). Then we try to prove to E analogous results already presented in the literature for rank 2 locally free sheaves associated to Y . The analogous properties are the stability criterion, splitting criterion, relations between the Chern classes of the vector bundle and the degree and genus of the associated curve and finally an equation which gives the dimension of the tangent space at a point E in the moduli space of rank 3 locally free sheaves in terms of the Chern classes of E.

From now on in this thesis we say that a curve Y in P3 satisfies the H-S correspondence if the curve satisfies the hypothesis of Theorem 2.6.

Theorem 3.1. Let Y be a curve in P3 which satisfies the H-S correspondence and let E be its associated rank 3 bundle with first Chern class c1. Then E is stable if and only if

1) c1 > 0

2) Y is not contained in a surface of degree lower than or equal to 2c1

3

3) (i) if 3 divides c1 then the induced cohomology map H0(2O( c1 3)) → H 0 (ωY(4− 2c1 3 )) is injective

(ii) if 3 divides c1 + 1 then the induced cohomology map H0(2O( c1− 2

3 )) →

H0(ωY(3 −

2c1− 1

3 )) is injective

(iii) if 3 divides c1 + 2 then the induced cohomology map H0(2O( c1− 1

3 )) →

H0(ωY(3 −

2c1− 2

3 )) is injective

Proof: Assume that 3 divides c1. From Theorem 1.6 we have the sequence:

0 → 2O → E → IY(c1) → 0

Twisting this exact sequence by O(−c1

3 ) we have: 0 → 2O → (−c1

3 ) → F → IY( 2c1

where F = Enorm = E(

−c1

3 ). Now E is stable if, and only if, H

0_{(F ) = H}0_(F∗

) = 0. Using the long exact sequence of cohomology associated to the last exact sequence we get that

H0(F ) = 0 is equivalent to

1) H0(−c1

3 ) = 0 which is equivalent to c1 > 0, 2) H0(IY(

2c1

3 )) = 0 which says that Y is not contained in any surface of degree 2c1

3 . Therefore the condition H0(F ) = 0 is equivalent to claims 1) and 2).

Moreover, dualizing the exact sequence: 0 → 2O(−c1

3 ) → F → IY( 2c1

3 ) → 0 we obtain the long exact sequence:

0 → O −2c 1 3  → F∗ → 2O _c 1 3  → Ext1_I Y _2c 1 3  , O  → 0

which divides in two short exact sequences

0 → O −2c 1 3  → F∗ → P → 0 0 → P → 2O _c 1 3  → Ext1_I Y _2c 1 3  , O  → 0

The long exact sequence of cohomology associated to the first exact sequence gives us

H0(F∗) = H0(P ) and the long exact sequence of cohomology associated by the second exact sequence is:

0 → H0(P ) → H0  2O c 1 3  → H0(E xt1  IY 2c 1 3  , O  → . . . We known that E xt1  IY _2c 1 3  , O  = ωY  4 −2c1 3 

Therefore, H0(F∗) = 0,if and only if, H0  2O _c 1 3  → H0_ω Y  4 −2c1 3  is injective. The cases 3|c1+ 1 and 3|c1+ 2 are similar.

3.2

Chern classes of rank 3 locally free sheaf and degree and genus

of its associated curve

In this section, we obtain a relation between the main invariants of a curve

Y ⊆ P3 which satisfies the Hartshorne-Serre correspondence and its associated rank 3 locally free sheaf.

This result is not original and it can be found in Vogelaar’s article ( See

(VOGELAAR, 1978)) but we give here a different proof. We use this result in the next

chapter to verify that the associated curve to a rank 3 instanton locally free sheaf with charge 2 (or 3) is a rational curve of degree 5 (or 6).

Theorem 3.2. Let Y be a curve in P3 satisfying the H-S correspondence and let E be the associated rank 3 locally free sheaf. Then

1) d = c2

2) c3− 4c2+ c1c2 = 2g − 2

where d and g are respectively the degree and genus of Y and c1, c2, c3 are the Chern classes of E.

Proof: Consider the exact sequence given by the Hartshorne-Serre Correspondence:

0 → 2O → E → IY(c1) → 0

Then we have PE(m) = 2PO(m) + PIY(m + c1) ∀m ∈ Z, where PE, PO and PIY are the

Hilbert polynomials of E, O, IY respectively. By the exact sequence:

0 → IY → O → OY → 0

we obtain PO(m) = POY(m) + POY(m). Then

PE(m) = 2PO(m) + PO(m + c1) − POY(m + c1)

By the Riemann-Roch formula, we get:

PE(m) = χ(E(m)) = (chE(m)) × td(P3))[H3_] Since ch(E) = 3 + c1H + 1 2(c1 2_{− 2c} 2)H2 + 1 6(c1 3 − 3c1c2+ 3c3)H3 and

ch(O(m)) = 1 + mH + m 2 2 H 2₊m3 6 H 3 then ch(E(m)) = ch(E)ch(O(m)) = 3+(c1+3m)H +( 1 2(c1 2_−2c 2)+mc1+ 3 2m 2_)H2₊ ((1 6c1 3_{− 3c} 1c2+ 3c3) + 1 2m(c1 2_{− 2c} 2) + m2_c 1 2 + m3 2 )H 3_. As td(P3) = 1 + 2H + 11 6 H 2_{+ H}3 _{we get} χ(E(m)) = 1 6(c1 3_{− 3c} 1c2+ 3c3) + 1 2m(c1 2_{− 2c} 2) + m2_c 1 2 + m3 2 + (c1 2_{− 2c} 2) + 2mc1+ 3m2+ 11 6 + 11 2 m + 3

On the other hand, using the equation PE(m) = 2PO(m) + PO(m + c1) − POY(m + c1) we

get χ(E(m)) = m + 3 3 ! + m + c1+ 3 3 ! − (m + c1)d − 1 + g

comparing the coefficients of the two expressions for χ(E(m)) of degree 0 and 1 we get d = c2 and c3− 4c2+ c1c2 = 2g − 2.

3.3

Relation between rank 2 and rank 3 vector bundles associated

a subcanonical curve

This section contains one of the main results of this thesis: the relation between rank 2 and rank 3 locally free sheaves associated to a given subcanonical curve in P3. As consequence, we obtain a criterion of decomposability of the rank 3 locally free sheaf associated to a globally complete intersection subcanonical curve in P3.

Proposition 3.3. Let Y ⊆ P3 be a curve that satisfies the H-S correspondence with

detN_{Y /P}3 = O_Y(k). Then the associated rank 2 and rank 3 bundles E and F respectively

are related by the following exact sequence

O → O → F → E → 0

0 //O i  α _// F 0 //2O (α,β) //F //IY(k) //0

where i : O → 2O is the inclusion on the first summand. Then we get the commutative diagram 0  0  0 //O i  α _// F //Coker(α) //0 0 //2O  (α,β) _// F //  IY(k) //0 O //O  //₀  //₀ 0 0

By Snake lemma we get an exact sequence

0 → O → coker(α) → IY(k) → 0

Set E = coker(η). Dualizing the short exact sequence above we get

0 → O(−k) → E∗ → O → Ext1_(I

Y(k), O) → Ext1(E, O) → Ext1(E, O) = 0

Using the Lemma 1.8, we get E xt1(IY(k), O) = ωY(4 − k) where ωY = ∧2NY /P3 ⊗ ω

P3.

Hence ωY(4 − k) = OY and the map O → ωY(4 − k) in the exact sequence is surjective.

This means that E xt1(E, O) = 0 which implies that E is a locally free sheaf. Therefore E is the rank 2 vector bundle associated to Y and the short exact sequence

0 //O α //F //E //0 proves the claim.

Proposition 3.4. Let Y in P3 be a curve that satisfies the H-S correspondence and let E be the associated rank 2 locally free sheaf. If F is a rank 3 locally free sheaf satisfying

O → O → F → E → 0

then F is the rank 3 vector bundle associated to Y via the H-S correspondence.

Proof: The short exact sequence above and the short exact sequence

0 → O → E → IY(k) → 0

given by H-S correspondence give us the commutative diagram

0  0  0  0 _// O  0 //O //F t //E s  //₀ F  u _// IY(k)  //₀ 0 0

where u = s ◦ t. By the Snake lemma we get the exact sequence

0 → ker(0) → ker(t) → ker(u) → coker(0) → coker(t) → coker(u) → 0

which gives the short exact sequence

0 → O → ker(u) → O → 0

As E xt1(O, O) = 0 we get ker(u) = 2O and the short exact sequence

0 → 2O → F → IY(k) → 0

| Proposition 3.5. Let Y ⊆ P3 a subcanonical and globally complete intersection curve that satisfies the H-S correspondence. Then the associated rank 3 locally free sheaf splits.

Proof: If F is the rank 3 and E the rank 2 locally free sheaves associated to Y , then by the above proposition there exists an exact sequence

0 → O → F → E → 0

As E is locally free, we have Ext1(E, O) = 0 and F = O ⊕ E. By the hypothesis in Y and the Corollary 2.1 we get that F splits.

3.4

Splitting criterion

Let Y ⊆ P3 be a curve which satisfies the hypothesis of Hartshorne-Serre correspondence (see 1.6) and let E be the rank 3 vector bundle associated to Y . In this section we use the concept of an Arithmetic Cohen-Macaulay curve to characterize precisely when the associated locally free sheaf splits.

Definition 3.6. _{Let Y be a curve in P}3 of pure dimension 1. We say that Y is Arithmetic Cohen-Macaulay if H1(IY(k)) = 0 for all k ∈ Z.( A.C.M for short)

Proposition 3.7. If Y is a globally complete intersection curve, then Y is A.C.M. Proof: If Y is a globally complete intersection curve, then Y = V (F1) ∩ V (F2). where F1 and F2 are homogeneous polinomials of degree d1 and d2 respectively. Then we have the exact sequence

0 → O(−d1− d2) → O(−d1) ⊕ O(−d2) → O → OY → 0 which give us the short exact sequence

0 → O(−d1− d2) → O(−d1) ⊕ O(−d2) → IY → 0 Twisting by O(k) we get

0 → O(−d1− d2+ k) → O(−d1+ k) ⊕ O(−d2+ k) → IY(k) → 0

By examining the corresponding long exact sequence of cohomology we obtain H1(IY(k)) =

0.

Definition 3.8. Let E be a locally free sheaf on Pn. Then we define its first module of cohomology as H∗1(E) = ⊕k∈ZH1(E(k)) .

Proposition 3.9. Let Y ⊆ P3 E be a curve which satisfies the H-S correspondence and let E be the associated rank 3 locally free sheaf. Then H∗1(E) = 0 if, and only if, Y is A.C.M. .

Proof: We twist the short exact sequence

0 → 2O → E → IY(c1) → 0

by O(k) and we get

0 → 2O(k) → E(k) → IY(c1+ k) → 0

By the corresponding long exact sequence of cohomology we get

H1(E(k)) ∼= H1(IY(k + c1))

from which the proposition follows.

The importance of the A.C.M property shall be clear with the following results. Definition 3.10. Let Y ⊂ P3 be a curve satisfying the H-S Correspondence and let E be the rank 3 associated locally free sheaf. Let n ∈ Z the smallest integer such that E∗(n) is

generated by a finite number of global sections. Then we can define the curve Z such that there exists a short exact sequence

0 → 2O → E∗(n) → IZ(3n − c1) → 0 We call the curve Z the dual curve of Y .

Theorem 3.11. Let Y ⊆ P3 be a curve satisfying the H-S correspondence, and let E be the associated rank 3 vector bundle. If Y is subcanonical and A.C.M, then E splits.

Proof: Consider the exact sequence

0 → 2O → E → IY(k) → 0

By dualizing, we have

0 → O(−k) → E∗ → 2O → ωY(4 − k) → 0

where E xt1(IY(k), O) = ωY(4 − k), by the lemma 1.8 and E xt1(E, O) = 0 because E is

locally free.

As Y is a subcanonical curve we have ωY(4 − k) = OY and then we get the exact sequence

Twisting by O(l), we have:

0 → O(l − k) → E∗(l) → 2O(l) → OY(l) → 0

We can break this exact sequence above in two short exact sequences

0 → O(l − k) → E∗(l) → K → 0

0 → K → 2O(l) → OY(l) → 0

We use the second short exact sequence to obtain the comutative diagram

0 //IY  //_O  //_OY ∼ =  //₀ 0 //K(−l) //2O //OY //0

where the first and second maps are natural injections. Thus we may use the Snake lemma to get the exact sequence

0 → IY → K(−l) → O → 0

As Ext1(O, IY) = H1(IY) = 0, we get

K = (IY ⊕ O)(l) = IY(l) ⊕ O(l)

From the long exact sequence of cohomology corresponding to the short exact sequence

0 → O(l − k) → E∗(l) → K → 0

we get H1(E∗(l)) = H1(K) = H1(IY(l) ⊕ O) = 0. Hence we have H∗1(E) = 0 and by

Serre duality we get H2_{(E)(k) = 0, ∀ k ∈ Z.} Moreover, from the exact sequence

and the corresponding long exact sequence of cohomology we get

H1(E(k)) = H1(IY(k + c1)) = 0

As H1(E(k)) = H2(E(k)) = 0, by Horrocks splitting criterion we get that E splits. In the last proof we used the following well known criterion:

Horrocks Splitting Criterion: _{Let E be a rank 3 vector bundle on P}3. Then E splits if and only H1(E(k)) = H2_{(E(k)) = 0 ∀ ∈ Z}

Proof: See (MIRó-ROIG, 1987) Lemma 1.3 pg 2.

Theorem 3.12. Let Y ⊆ P3 be a curve satisfying the H-S correspondence. Then the associated locally free sheaf E splits if and only if Y and its dual curve are A.C.M.

Proof: By the Horrocks splitting criterion, E splits if and only if H1(E(k)) = 0 and

H2_{(E(k)) = 0 ∀ k ∈ Z. The claim H}1_{(E(k)) = 0 ∀ k ∈ Z occurs when the associated curve} Y is A,C,M, by the Theorem 3.9. The claim H2_{(E(k)) = 0 ∀ k ∈ Z by Serre duality is} equivalent to H1(E∗_{(k)) = 0 ∀ k ∈ Z. Then by using again the Theorem}3.9 we conclude that the required claim ocurs when the dual curve is A.C.M.

3.5

Expected dimension of moduli space of rank 3 vector bundles

with Chern classes c

, c

, c

In this section we prove an useful property to compute the dimension of the tangent space of an irreducible component in the moduli space of stable rank 3 locally free sheaves of fixed Chern classes. We shall use this property to compute dim Ext1(E, E), where E is the rank 3 instanton locally free sheaf with charge 2. We will construct a example of a rank 3 instanton locally free sheaf of charge 2 whose dimension of Ext2(E, E), is zero which yields dim Ext1(E, E) = 16. From this we obtain a example of an instanton locally free sheaf that is a nonsingular point in the moduli space of stable rank 3 locally free sheaves with given Chern classes (c1, c2.c3) = (0, 2, 0).

Proposition 3.13. Let E be a rank 3 locally free sheaf with Chern classes c1, c2, c3. Then χ(E ⊗ E∗) = 4c12− 12c2+ 9

Proof: Let E be a rank 3 vector bundle and consider L = c1(OP3(1)), the class of a

hyperplane in the Chow ring of P3, then

ch(E) = 3 + c1L + 1 2(c1 2_{− 2c} 2)L2+ 1 6(c1 3_{− 3c} 1c2+ 3c3)L3

and ch(E∗) = 3 − c1L + 1 2(c1 2_{− 2c} 2)L2+ 1 6(−c1 3_{+ 3c} 1c2− 3c3)L3 Hence ch(E ⊗ E∗) = ch(E)ch(E∗) = 9 − 2(c12 − 3c2)L2

and using Grothendieck-Riemann-Roch theorem we have:

χ(E ⊗ E∗) = (ch(E ⊗ E∗_{) × td(P}3))_[L3_] = 4c₁2− 12c₂+ 9

Proposition 3.14. Let E be a stable rank 3 vector bundle on P3. Then E ⊗ E∗ is semistable.

Proof: See (HUYBRECHTS, 2010) Lemma 1.5.10 pg 23

Proposition 3.15. Let E be a stable vector bundle on Pn, then dim Hom(E, E) = 1

Proof: See (OKONEK et al., 1980) Theorem 1.2.9 p 87

Proposition 3.16. Let E be a stable rank 3 vector bundle on P3 with Chern classes c1, c2, c3. Then :

dim Ext1(E, E) − dim Ext2(E, E) = −4c12 + 12c2− 8

Proof: If E is stable, then dim(Ext0(E, E)) = dim H0(Hom(E, E)) = 1. By Serre duality and the semistability of E ⊗ E∗. we obtain:

Ext3(E, E) = Hom(E, E∗(−4))∗ = 0 Then

4c12− 12c2+ 9 = 1 − dim Ext1(E, E) + dim Ext2(E, E)

4 The moduli space of rank 3 instanton

bun-dles without global sections of charge 2

4.1

Rank 3 instanton sheaves without global sections and rank 2

stable reflexive n sheaves

The main goal of this section is to relate rank 3 locally free instanton sheaves of charge 2 without global sections with stable rank 2 reflexive sheaves with Chern classes (c1, c2, c3) = (1, 3, 3). Then we use this result to prove that the family of rank 3 instanton

bundles without global sections of charge 2 has dimension 16.

In order to get this main result we need to prove several propositions starting with the one which calculates the spectra of an instanton rank 3 bundle with charge 2. Proposition 4.1. _{Let E be a semistable rank 3 vector bundle on P}3 and n be its second Chern class. Then E is an instanton bundle if and only if kE = (0, . . . , 0)

| {z }

n .

Proof: Let E be a rank 3 semistable vector bundle of charge n on P3 and let kE =

(k1, . . . , kn) be its spectra. If E is an instanton bundle of charge n on P3then h1(E(−2)) = 0

which implies by Theorem1.26

h1(E(−2)) = h0(⊕n_i=1O_P1(k_i− 1)) =

n

X

i=1

h0(O_P1(k_i− 1))

Hence ki ≤ 0 ∀i ∈ {1, . . . , n}. By the theorem1.27 k1+ . . . + kn= 0( since c1(E) = 0).

As for any spectra we have k1 ≤ k2 ≤ . . . ≤ kn, then kE = (0, . . . , 0)

| {z }

n

.

Reciprocally if E is a semistable vector bundle with spectra kE = (0, . . . , 0)

| {z }

n

. Then 0 = h0(E(−1)) = h0(E∗(−1)) since E is semistable. By Serre duality, we get

h3(E(−3))=0. Moroever since that kE = (0, . . . , 0)

| {z }

n

, we get

(i) h1(E(−2)) = h0(⊕n_i=1O_P1(−1)) = 0

(ii) h2(E(−2)) = h1(⊕n_i=1O_P1(−1)) = h0(⊕n_i=1O

P1(−1)) = 0

The next theorem associates a reflexive rank 2 stable linear sheaf N of Chern classes (1, 3, 3) to an rank 3 instanton bundle E of change 2 without global sections. But before this we need the following

Proposition 4.2. Let E be a rank 3 instanton bundle on P3. Then given m ∈ Z, we have: χ(E(m)) = m

3

2 + 3m

2_{+ (}11

2 − n)m + (3 − 2n)

where n is the charge of E.

Proof: Straightforward calculation replacing (c1, c2, c3) = (0, 2, 0) in the formula χ(E(m)) = 1 6(c1 3_{− 3c} 1c2+ 3c3) + 1 2m(c1 2_{− 2c} 2) + m2_c 1 2 + m3 2 + (c1 2_{− 2c} 2) + 2mc1+ 3m2+ 11 6 + 11 2 m + 3

given by Theorem 3.2 gives us the proposition. .

Theorem 4.3. Let E be a rank 3 instanton bundle of charge 2 on P3 such that H0(E) = 0.

Then H0(E(1)) 6= 0 and given s ∈ H0(E(1)) non trivial, we get a reflexive rank 2 stable

linear sheaf N on P3 with Chern classes (1, 3, 3) such that H0(N ) = 0. The section s

induces an exact sequence

0 → O(−1)→ E → N → 0s

Proof: By Proposition 4.2 taking m = 1, we get χ(E(1)) = 6. Then

6 = χ(E(1)) = h0(E(1)) − h1(E(1)) + h2(E(1)) − h3(E(1))

By Proposition 4.1, kE = (0, 0) and hence

h2(E(1)) = 2h1(O_P1(2)) = 2h0(O

P1(−4)) = 0

By Serre duality,

h3(E(1)) = h0(E∗(−5)) = 0

The last equality comes from the fact that as E is a stable sheaf so is E∗ and then since

c1(E∗) = 0 we get H0(E∗) = 0 which yields H0(E∗(−5)) = 0. Then

which yields h0(E(1)) ≤ 6. Hence we can construct a torsion free sheaf N such that

0 → O(−1)→ E → N → 0s

where s in a non trivial global section. Recall that N is torsion free because s ∈ H0(E(1)) is a global section and H0(E) = 0. The cohomology sequence associated do the short enact sequence above gives us

H0(N ) = H0(E) = 0 Dualizing the above exact sequence, we get

0 → N∗ → E∗ → O(1) → . . .

Breaking the exact sequence and using cohomology of the short exact sequence, we have

H0(N∗) = 0 which implies that N is stable. Using once more the cohomology sequence we get

(i) H0(N (−1)) ∼= H0(E(−1)) = 0

(ii) 0 = H1(E(−2)) → H1(N (−2)) → H2(O(−3)) = 0

(iii) 0 = H2(E(−2)) → H2(N (−2)) → H3(O(−3)) = H0(O(−1)) = 0 (iv) 0 = H3(E(−3)) → H3(N (−3)) → H4(O(−4)) = 0

Thus N is a linear sheaf.

Thus the sheaf N constructed in the above proposition is the cohomology of a monad of the form

0 → 3O(−1) → 7O → 2O(1) → 0

Let us compute the Chern classes of N using this monad. In fact, consider more generally the cohomology N of a monad of the form

0 ← (n + 1)O(−1) → (2n + 3)O → nO(1) → 0 This monad gives us two exact sequences

0 → (n + 1)O(−1) → K → E → 0 Then using the Chern polynomial, we conclude that

ct(N ) = ( 1 1 − t) n+1₍ 1 1 + t) n = ( 1 1 − t)( 1 1 + t2) n = ( 1 1 − t)(1 + nt 2₎ = (1 + t + t2+ t3)(1 + nt2) = 1 + t + (n + 1)t2+ (n + 1)t3 (4.1)

Thus N has Chern classes (1, n + 1, n + 1) and in particular in the case n = 2 we have (1, 3, 3) as its Chern classes.

The next goal is is to prove that N is indeed a reflexive sheaf. For this purpose, we need to study the curve associated to a rank 3 stable instanton bundle E of charge 2 by the Hartshorne-Serre correspondence.

Proposition 4.4. Consider a rank 3 locally free instanton sheaf E on P3 of charge n, without global sections and suppose that H0(E(1)) ≥ 2. Then we can write E(1) as the

middle term of an exact sequence

0 → 2O → E(1) → IY(3) → 0

where Y is a rational curve in P3 of degree n + 3.

Proof: Indeed, as h0(E(1)) ≥ 2 we get two linearly independent sections of E(1) which determine an exact sequence

0 → 2O → E(1) → IY(3) → 0

where Y is a locally complete intersection curve in P3. The Chern classes of

E(1) are (3, n + 3, n + 1). Then using the Theorem 3.2, we have:

d = n + 3

n + 1 − 4(n + 3) + 3(n + 1) = 2g − 2

Corollary 4.5. Let E be a rank 3 instanton bundle of charge 2 without global sections

such that H0(E(1)) ≥ 2. Then the associated curve Y to E(1) by the Hartshorne-Serre

correspondence is a rational curve of degree 5.

Now we consider E to be a rank 3 instanton bundle of charge 2 without global sections such that H0(E(1)) ≥ 2 and we prove that the associated rank 2 linear sheaf N is indeed reflexive. In this case we know that

6 = χ(E(1)) = h0(E(1)) − h1(E(1)) so h0_{(E(1)) > 6.}

Then considering two general global sections s and σ of E(1) we can construct an exact sequence of the form

0 → 2O → E(1) → IY(3) → 0

where Y is the curve given by the points in P3 where s and σ are linearly independent sections and c1(E(1)) = 3. Then we can construct the following commutative diagram

0  0  0 //O i  id _// O s  0 //2O π  (s,σ)_// E(1) //  IY(3) //0 O  l _// N (1)  0 0

where the exact sequence

0 → O → E(1) → N (1) → 0 is obtained as before.

Applying the Snake Lemma, we have Coker(l) ∼= IY(3) and l is injective. Hence we get

the short exact sequence

Then N (1) is reflexive by Proposition1.11. Indeed, we Know that Y is a locally complete intersection and Cohen-Macaulay curve and besides ωY(4 − c1) is generated by a global

section except a finite number of points(Because Y is a rational curve). Then N is an reflexive sheaf and stable as promised.

Now we want to reverse the construction of the last proposition and given a rank 2 reflexive and stable sheaf N in P3 with Chern classes (1, 3, 3) to get a rank 3 stable instanton sheaf (and therefore without global sections) and a section s ∈ H0(E(1)) such that there exists an exact sequence

0 → O(−1) → E → N → 0 Before proving the the above assertion we need a lemma

Lemma 4.6. Let N be a semistable reflexive rank 2 linear sheaf in P3 with Chern classes

(c1, c2, c3) = (1, n + 1, n + 1). Then

kNnorm = (−1. . . . , −1)

.

Proof: If

kNnorm = (k1. . . . , kn+1)

then k1+ . . . + kn+1= −n − 1. Hence k1 ≤ −1 and kn+1 ≥ −1. If N is linear then

0 = h1(N (−2)) = h0(⊕n+1_i=1O(ki− 1)) = n+1

X

i=1

h0(O(ki− 1))

. So kn+1 ≤ −1, hence kn+1 = −1. Thus k1+ . . . + kn = −n and k1 = . . . = kn+1 = −1

and the lemma follows.

Theorem 4.7. Let N be a rank 2 stable reflexive linear sheaf with Chern classes (c1, c2, c3) =

(1, 3, 3) Then there exists a short exact sequence

0 → O(−1) → E → N → 0

Where E is a rank 3 stable instanton sheaf of charge 2.

Proof By the short exact sequence above we should look at the group Ext1(N, O(−1)). First we prove that this group is nontrivial. We have