Universidade Estadual de Campinas
Instituto de Matemática, Estatística e Computação Científica
Brady Miliwska Ali Medina
The Riemann–Roch Theorem and different ways to generalize the Weierstrass Semigroup
O Teorema de Riemann-Roch e diferentes formas de generalizar o Semigrupo de Weierstrass
CAMPINAS 2020 Brady Miliwska Ali Medina
The Riemann–Roch Theorem and different ways to generalize the Weierstrass Semigroup
O Teorema de Riemann-Roch e diferentes formas de generalizar o Semigrupo de Weierstrass
Dissertação apresentada ao Instituto de Matemática, Estatística e Computação Científica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestra em Matemática.
Dissertation presented to the Institute of Mathematics, Statistics and Scientific Computing of the University of Campinas in partial fulfillment of the requirements for the degree of Master in Mathematics.
Orientador: Marcos Benevenuto Jardim
Este exemplar corresponde à versão final da dissertação defendida pela aluna Brady Miliwska Ali Medina e orientada pelo Prof. Dr. Marcos Benevenuto Jardim.
Campinas 2020 Ficha catalográfica Universidade Estadual de Campinas Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Ali Medina, Brady Miliwska, 1997- AL41r AliThe Riemann-Roch theorem and different ways to generalize the Weierstrass semigroup / Brady Miliwska Ali Medina. – Campinas, SP : [s.n.], 2020.
AliOrientador: Marcos Benevenuto Jardim. AliDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Ali1. Riemmann-Roch, Teorema de. 2. Curvas algébricas. 3. Fibrados vetoriais. 4. Weierstrass, Semigrupos de. I. Jardim, Marcos Benevenuto, 1973-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: O teorema de Riemann-Roch e diferentes formas de generalizar o semigrupo de Weierstrass Palavras-chave em inglês: Riemann-Roch theorem Algebraic curves Vector bundles Weierstrass semigroups Área de concentração: Matemática Titulação: Mestra em Matemática Banca examinadora: Marcos Benevenuto Jardim Fernando Eduardo Torres Orihuela Ethan Guy Cotterill Data de defesa: 06-03-2020 Programa de Pós-Graduação: Matemática
Identificação e informações acadêmicas do(a) aluno(a) - ORCID do autor: https://orcid.org/0000-0001-9612-497X - Currículo Lattes do autor: http://lattes.cnpq.br/5226841950913825
Powered by TCPDF (www.tcpdf.org) Dissertação de Mestrado defendida em 06 de março de 2020 e aprovada
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). MARCOS BENEVENUTO JARDIM
Prof(a). Dr(a). FERNANDO EDUARDO TORRES ORIHUELA
Prof(a). Dr(a). ETHAN GUY COTTERILL
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.
To my family. Acknowledgments
First, I want to thank God.
I am also grateful to my parents Armando and Rita for always supporting me in every way. I would not have been able to study a master’s degree if it weren’t for the incredible guide you always gave me. In addition, I want to thank my two little sisters Keyka and Amy for their emotional support and for making my life happier than it is.
I would like to express my gratitude to my advisor Prof. Marcos Jardim for all the opportunities offered, his time, his patience and his support. He is certainly an example to follow bothasa professor and person. It is a great honor to work with him.
I am grateful to Prof. Mahendra Panthee and Prof. Fernando Torres for their support during this master’s. I also want to thank Prof. Lucas Catao for his cordial welcome when I came to Brazil and his support.
I also want to thank Juan P. for his emotional support and all my friends for the timeshare.
I would like to recognize the invaluable assistance of the staff of the Secretariat of Graduate Studies, they helped me a lot with the paperwork.
This study was financed by Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, through the process 2018/12888-0, from 01/11/2018 to 29/02/2020. Also, this study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) - Finance Code 001. I want to thank both FAPESP, and CAPES for their financial support. Resumo O objetivo desta tese é provar o Teorema de Riemann–Roch para uma curva projetiva suave e dar diferentes formas de generalizar o conceito de um semigrupo de Weierstrass 퐻푃 de um ponto P em X. Começamos por definir o semigrupo de Weierstrass 퐻(퐷) de um divisor 퐷 e obtemos que o maior número do conjunto de lacunas é inferior a 2g. Depois, definimos o semigrupo de Weierstrass 퐻(퐸, 푃 ) de um divisor 퐸 em relação ao ponto 푃 e obtemos que a cardinalidade do conjunto de lacunas é 푙(퐾푋 − 퐸). Em seguida, definimos o semigrupo de Weierstrass 퐻(퐸, 퐷) de um divisor 퐸 com respeito a 퐷 e obtemos que o número máximo do conjunto de lacunas é inferior a 2푔 − deg(퐸)/deg(퐷). Finalmente, definimos o conjunto de Weierstrass 푆(ℱ, 푃 ) de um fibrado vetorial ℱ com respeito a 푃 e provamos que é um 퐻푃 -ideal relativo. Além disso, se ℱ for semiestável, então provamos que o número máximo do conjunto de lacunas é inferior a 2푔 − deg(ℱ)/rk(ℱ).
Palavras-chave: Riemman-Roch, Curvas Algebricas, Fibrados Vetoriais, Semigrupo de Weierstrass. Abstract The objective of this thesis is to prove the Riemann–Roch Theorem for a smooth projective curve 푋, and to give different ways to generalize the concept of a Weierstrass semigroup 퐻푃 of a point 푃 in 푋. We begin by defining the Weierstrass semigroup 퐻(퐷) of a divisor 퐷 and we get that the largest gap is less than 2푔. Then, we define the Weierstrass semigroup 퐻(퐸, 푃 ) of a divisor 퐸 with respect to a point 푃 and we obtain that the cardinality of the set of gaps is 푙(퐾푋 − 퐸). Afterwards, we define the Weierstrass semigroup 퐻(퐸, 퐷) of a divisor 퐸 with respect to 퐷 and we have that the largest gap is less than 2푔 − deg(퐸)/deg(퐷). Finally, we define the Weierstrass set 푆(ℱ, 푃 ) of a vector bundle ℱ with respect to 푃 and we prove that 푆(ℱ, 푃 ) is an 퐻푃 -ideal. Furthermore, if ℱ is semistable then we prove that the largest gap is less than 2푔 − deg(ℱ)/rk(ℱ).
Keywords: Riemman-Roch, Algebraic Curves, Vector Bundles, Weierstrass Semigroup. Contents
Introduction 10
1 Preliminaries 13 1.1 Vector Bundles and Locally free sheaves ...... 13 1.2 Divisors ...... 18 1.2.1 Weil Divisors ...... 18 1.2.2 Cartier Divisors ...... 20 1.2.3 Correspondence between Weil Divisors and Cartier Divisors ...... 21 1.2.4 The invertible sheaf associated to a divisor ...... 21 1.2.5 Canonical Class ...... 24 1.3 Stable Bundles ...... 26
2 Riemann–Roch Theorem 27 2.1 Čech Cohomology ...... 27 2.2 Riemann–Roch Theorem ...... 29
3 The Weierstrass Semigroup 37 3.1 The Weierstrass Semigroup of a point P ...... 37 3.2 The Weierstrass Semigroup of a divisor 퐷 ...... 41 3.3 The Weierstrass Semigroup of a divisor E with respect to a point P ...... 43 3.4 The Weierstrass Semigroup of a divisor 퐸 with respect to a divisor 퐷 ...... 46 3.5 The Weierstrass set of ℱ with respect to a point 푃 ...... 49
Bibliography 51 10
Introduction
∑︀푟 ∑︀푠 Let 푋 be a nonsingular projective curve of genus 푔, and let 퐷 = 푖=1 푛푖푃푖 − 푗=1 푚푗푄푗 be a divisor on 푋 with 푛푖 > 0 and 푚푗 > 0. The Riemann–Roch Theorem computes the dimension of the linear space L(퐷) = {푓 ∈ 푘(푋)*|퐷 + 푑푖푣(푓) ≥ 0} ∪ {0}. * which is the set of all rational functions 푓 ∈ 푘(푋) such that 푓 has zeros of order ≥ 푚푗 at 푄푗, for 푗 = 1, ..., 푠, and 푓 may have poles only at 푃1, ..., 푃푟 with pole order at 푃푖 being bounded by 푛푖, 푖 = 1, ..., 푟. Riemann stated that:
dim L(퐷) ≥ deg(퐷) + 1 − 푔 and Roch provided the error term
dim L(퐷) − dim L(퐾푋 − 퐷) = deg(퐷) + 1 − 푔 giving origing to the Riemann–Roch Theorem. This theorem relates the zeros and poles of a function on a curve of genus 푔 and it is an example of how a mathematical result can stay alive due to its numerous applications. One of these applications is the Gap Theorem or the so-called Lückensatz. It is not clear when Weierstrass proved this, however, it was probably in the early 1860s, see [5, p. 38]. The Gap Theorem says that for each 푃 ∈ 푋 there are exactly 푔 integers 푛 such that there exist no rational function 푓 ∈ 푘(푋) having a pole at 푃 of multiplicity 푛. These integers 푛 are called gaps. In other words, the cardinality of the set
+ 퐺푃 := {푛 ∈ Z : there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 } is the genus 푔 of 푋. The complement of 퐺푃 is denoted by 퐻푃 and it has the algebraic structure of a numerical semigroup which is a set that contains the element zero 0, its complement with respect to the set of positive integers Z+ is finite, and if 푥, 푦 belongs to this set then 푥 + 푦 is also in this set. The set 퐻푃 is particularly called the semigroup of Weierstrass of 푃 . The study of the semigroup of Weierstrass 퐻푃 is important because it contributed to the understanding of the geometry of algebraic curves. Moreover, many applications have been found to code theory, see [4, 13]. Arbarello, Cornalba, Griffiths and Harris introduced the concept of the Weierstrass semigroup at two points in [7, p. 365]. In addition, they found a lower bound for the cardinality of all the gaps on two points of an algebraic curve 푋. In [11] Kim obtained formulas for the cardinality of all the gaps on two points in 푋. Also, he obtained lower and upper bounds of such sets. Introduction 11
Subsequently, Homma and Kim improved these results in [9, 10]. The concept of the Weierstrass semigroup at 푛 points has been generalized in at least two different ways, see Beelen, Tutas [1] and Carvalho [3]. In chapter 3 of this thesis wewillgive different ways to generalize the concept of the Weierstrass semigroup and some results inthe cardinality of the set of gaps.
The structure of this thesis is as follows:
Chapter I In this chapter we will explain some concepts necessary for the understanding of this thesis. We start by defining vector bundles over a variety 푋 and locally free sheaves. Then, we prove that there is a one to one correspondence between these two notions. Next, we define Weil divisors and Cartier divisors and we also prove that if 푋 is a smooth algebraic variety then there exist a one to one correspondence between Weil divisors and Cartier divisors. After that, we define the sheaf ℒ퐷 associated to a divisor 퐷. We also define the vector space L(퐷) and prove that the set of global sections Γ(푋, ℒ퐷) of the sheaf ℒ퐷 is isomorphic to L(퐷). Later, we define the Canonical sheaf 휔푋 of 푋 and the associated divisor is called the canonical divisor 퐾푋 . To finish, we define (semi)stable vector bundles and we give some properties ofthem.
Chapter II In this chapter we define the 푝th Cěch cohomology group and prove the Riemann–Roch Theorem for vector bundles on a nonsingular projective curve 푋. Furthermore, we give some properties of semistable vector bundles.
Chapter III In this chapter we start by defining the Weierstrass semigroup 퐻푃 of a point 푃 ∈ 푋, and we also prove the Gap Theorem.
Next, we define the Weierstrass semigroup of a divisor 퐷 as
+ 퐻(퐷) := {푛 ∈ Z | there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷}.
Note that if we take 퐷 = 푃 this definition is equal to the definition of 퐻푃 . We prove in Proposition 3.2.3 that the largest gap is less than 2푔. Moreover, we prove in Theorem 3.2.4 that this is a numerical semigroup.
Afterwards, we define the Weierstrass semigroup of a divisor 퐸 with respect to a point 푃 as + 퐻(퐸, 푃 ) := {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛푃 }. Note that if we take 퐸 = 0 this definition is equal to the definition of 퐻(퐷). We prove in Proposition 3.3.3 that the largest gap is less than 2푔 − deg(퐸) and in Theorem 3.3.6 that if 퐸 is an effective, principal divisor, then 퐻(퐸, 푃 ) is a numerical semigroup. We also prove in Theorem 3.3.5 that the cardinality of the set of gaps 퐺(퐸, 푃 ) is exactly 푙(퐾푋 − 퐸) which is a theorem analogous to the Gap Theorem. Note that if 퐸 = 0 we have that 푙(퐾푋 − 퐸) = 푙(0) − deg(0) − 1 + 푔 = 푔 which is equal to the result obtained from the Gap Theorem. Introduction 12
Later, we define the Weierstrass semigroup of a divisor 퐸 with respect to a divisor 퐷 as
+ 퐻(퐸, 퐷) := {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛퐷}.
Note that if we take 퐷 = 푃 this definition is equal to the definition of 퐻(퐸, 푃 ) and if we take 퐸 = 0 this definition is equal to the definition of 퐻(퐷). We obtained in Proposition 3.4.3 that the largest gap of 퐺(퐸, 퐷) is less than 2푔−deg(퐸)/deg(퐷). Also we proved in Theorem 3.4.4 that 퐻(퐸, 퐷) is a numerical semigroup in the case that 퐸 is an effective, principal divisor and 퐷 is an effective divisor.
To finish we define the Weierstrass set
+ 0 0 푆(ℱ, 푃 ) := {푛 ∈ Z |ℎ (ℱ(푛 − 1)푃 ) < ℎ (ℱ(푛푃 ))} where ℱ(푛푃 ) := ℱ ⊗ 풪푋 (푛푃 ). We prove in Theorem 3.5.3 that 푆(ℱ, 푃 ) is an 퐻푃 -relative ideal, and in Theorem 3.5.4 that if ℱ is a semistable bundle then the largest gap is less than ∼ 2푔 − deg(ℱ)/rk(ℱ). In the case when ℱ is a line bundle we have that ℱ = 풪푋 (퐸) for some divisor 퐸, and the Definition of the Weierstrass set 푆(ℱ, 푃 ) is the same as the Definition of the Weierstrass semigroup of a divisor 퐸 with respect to 푃 . Also in this case, the largest gap is less than 2푔 − deg(퐸) as expected from Proposition 3.3.3. 13
Chapter 1
Preliminaries
In this chapter we will give the basic definitions needed for the development of this thesis. The lay out of this chapter is as follows. In section 1.1 we will give some basic facts about vector bundles and locally free sheaves. We will also see that there is a one-to-one correspondence between these notions. In section 1.2 we will define Weil divisors and Cartier divisors. Moreover, we will establish a one-to-one correspondence between Weil divisors and Cartier divisors. After that, we will prove an important correspondence which states that to every divisor we can associate an invertible sheaf and vice versa. To finish, we will define the canonical class divisor. In section 1.3 we will give some basic facts about (semi)stable vector bundles. In this thesis 푘 will always be an algebraic closed field.
1.1 Vector Bundles and Locally free sheaves
The study of Vector Bundles is important because of its connections with other branches of mathematics and its applications to physics. Definition 1.1.1. A vector bundle over a variety 푋 is a variety 퐸, and a morphism 휋 : 퐸 → 푋 satisfying the following conditions
1. There exists an open cover {푈푖}푖∈퐼 of 푋. 2. There exists a 푘-vector space 푉 , such that
−1 휓푖 : 휋 (푈푖) → 푈푖 × 푉
−1 is an isomorphism, and 휋 ∘ 휓푖 is the projection in the first coordinate. −1 3. The automorphism 휓푖,푗 = 휓푖 ∘ 휓푗 |푈푖∩푈푗 : (푈푖 ∩ 푈푗) × 푉 → (푈푖 ∩ 푈푗) × 푉 is a linear isomorphism for all 푖, 푗, that is, there exists an invertible matrix 퐴푖푗 such that
휓푖,푗(푥, 푣) = (푥, 퐴푖푗(푥) · 푣).
The matrixes 퐴푖,푗 are called transition matrixes if dim 푉 > 1, and transiction functions if dim 푉 = 1. CHAPTER 1. PRELIMINARIES 14
−1 A fiber 퐸푥 of 퐸 is defined by 퐸푥 := 휋 (푥), for 푥 ∈ 푋. If 푋 is connected, the rank of a vector bundle is well defined and is given by rk(퐸) = dim 퐸푥 for every 푥 ∈ 푋.
Example 1.1.2. Let 푋 be a variety, then the trivial vector bundle is given by
휋 : 푋 × 푉 → 푋 (푝, 휆) → 푝
Definition 1.1.3. Let 퐸 be a vector bundle over a variety 푋. If rk(퐸) = 1, then 퐸 is called a line bundle. The next proposition help us to construct new vector bundles. This proposition can be found in [12, p. 8].
Proposition 1.1.4. Let 퐸 be a vector bundle over an algebraic variety 푋. Let {푈푖} be an open cover of 푋, and 퐴푖,푗 the corresponding matrixes of 퐸 to the covering. Then the matrixes 퐴푖,푗 satisfy the following conditions
(i) 퐴푖푖 is the identity matrix.
(ii) 퐴푖푗퐴푗푘 = 퐴푖푘.
Reciprocally, given a set of matrixes {퐴푖,푗}푖,푗∈퐼 satisfying (i) and (ii) there is a vector bundle over 푋 whose transition matrixes are the given matrixes. ⋃︀ Example 1.1.5. If 푋 = 푈훼 is a cover in which the vector bundles 퐸 and 퐹 are defined by the transition matrixes 퐶훼훽 and 퐷훼훽, then [︃ ]︃ 퐶훼훽 0 1. 퐸 ⊕ 퐹 is a vector bundle defined by the transition matrix . 0 퐷훼훽
2. 퐸 ⊗ 퐹 is a vector bundle defined by the transition matrix 퐶훼훽 ⊗ 퐷훼훽. ∨ 푇 −1 3. 퐸 is a vector bundle, called the dual bundle, defined by the transition matrix (퐶훼훽) . ⋀︀푝 ⋀︀푝 4. 퐸 is a vector bundle defined by the transition matrix 퐶훼훽. If 푝 = rk(퐸), we write ⋀︀푝 퐸 = 푑푒푡(퐸).This is a rank 1 vector bundle, called the determinant line bundle of 퐸. The bundle 푑푒푡(퐸) is defined by the 1 × 1 matrixes 푑푒푡(퐶훼훽). Also, we can see that rk(퐸 ⊕ 퐹 ) = rk(퐸) + rk(퐹 ) and rk(퐸 ⊗ 퐹 ) = rk(퐸) · rk(퐹 ). Definition 1.1.6. Let 휋 : 퐸 → 푋, and 휋′ : 퐹 → 푋 be two vector bundles, a map 푓 : 퐸 → 퐹 is called a morphism of vector bundles if the diagram
푓 퐸 퐹
휋 휋′ 푋
commutes, and for each point 푥 ∈ 푋 the map 푓|퐸푥 : 퐸푥 → 퐹푥 is linear. CHAPTER 1. PRELIMINARIES 15
Definition 1.1.7. A (global) section of a vector bundle 휋 : 퐸 → 푋 is a morphism 푠 : 푋 → 퐸 such that 휋 ∘ 푠 = 푖푑 on 푋. Let 푈 ⊂ 푋 be an open subset, a morphism 푠 : 푈 → 퐸 such that 휋 ∘ 푠|푈 = 푖푑|푈 is called a local section. Let 0푥 be the zero vector in 퐸푥, then 푠(푥) = 0푥 is called the zero section of 퐸. The set of sections of a vector bundle 퐸 is defined as
ℒ(퐸) := {푠 : 푋 → 퐸|휋 ∘ 푠 = 푖푑 on 푋}.
Example 1.1.8. The sections of the trivial rank 1 bundle 휋 : 푋 × 푘 → 푋 are the morphisms 푝 → (푝, 푓(푝)). This is, given a section of the trivial rank 1 bundle is the same as giving an 1 element 푓 : 푋 → A of the ring of regular functions 풪푋 (푋). Therefore, ℒ(푋 × 푘) = 풪푋 (푋). Definition 1.1.9. A sequence of morphisms of vector bundles over a variety 푋
푓 푔 0 → 퐸′ −→ 퐸 −→ 퐸′′ → 0 is an exact sequence of vector bundles if
′ 푓푥 푔푥 ′′ 0 → 퐸푥 −→ 퐸푥 −→ 퐸푥 → 0 is an exact sequence of vector spaces for all 푥 ∈ 푋. 퐸′ is called a subbundle of 퐸, and 퐸′′ is called a factor bundle of 퐸. Now, we can define the quotient bundle 퐸/퐹 . Definition 1.1.10. The quotient bundle 퐸/퐹 of a vector bundle 퐸 by a subbundle 퐹 ⊂ 퐸 is defined as ⋃︁ 퐸/퐹 = 퐸푥/퐹푥. 푥∈푋
Also, if 0 → 퐸′ → 퐸 → 퐸′′ → 0 is an exact sequence, we say that 퐸 is an extension of 퐸′′ by 퐸′. 푓 푔 Proposition 1.1.11. Let 0 → 퐸′ −→ 퐸 −→ 퐸′′ → 0 be an exact sequence of vector bundles. Then, rk(퐸) = rk(퐸′) + rk(퐸′′).
Proof. Since 푓 푔 0 → 퐸′ −→ 퐸 −→ 퐸′′ → 0 is an exact sequence of vector bundles, we have that
′ 푓푥 푔푥 ′′ 0 → 퐸푥 −→ 퐸푥 −→ 퐸푥 → 0 is an exact sequence of vector spaces for all 푥 ∈ 푋. The dimension of a vector space is an additive function and thus the result follows.
Next, we will define a locally free sheave and we will see that locally free sheaves areinaone to one correspondence with vector bundles. Let 푋 be an algebraic variety, from now on 풪푋 will denote the sheaf of regular functions, and 풦푋 the sheaf of rational functions. CHAPTER 1. PRELIMINARIES 16
Definition 1.1.12. A sheaf of 풪푋 -modules ℱ is called locally free of rank 푛 if there is an ∼ ⊕푛 open cover {푈푖} of 푋 such that ℱ|푈푖 = 풪푈푖 for all 푖. Definition 1.1.13. A locally free sheaf of rank one is called an invertible sheaf.
Example 1.1.14. Let ℱ and 풢 be locally free sheaves of 풪푋 -modules of ranks 푟 and 푠 respectively. We have that 1. The tensor product ℱ ⊗ 풢 defined as the sheaf associated to the presheaf
푈 → ℱ(푈) ⊗푂푋 (푈) 풢(푈)
is also a locally free sheaf of 풪푋 -modules of rank 푟푠. 2. The 푛th exterior product of ℱ defined as the sheaf associated to the presheaf
푟 ⋀︁ 푈 → ℱ(푈)
is a locally free sheaf of 풪푋 -modules.
Proposition 1.1.15. The set of sections ℒ퐸 of a vector bundle 퐸 of rank 푟 is a locally free sheaf of 풪푋 -modules.
Proof. For any open set 푈 ⊂ 푋, define
ℒ퐸(푈) := {푠|푈 : 푈 → 퐸 such that 휋 ∘ 푠|푈 = 푖푑푈 }.
We can see that ℒ퐸(푈) is a sheaf of 풪푋 -modules. In fact, if 푠1, 푠2 ∈ ℒ퐸(푈), then 푠1 + 푠2 ∈ ℒ퐸(푈). Also, if 푓 ∈ 풪푋 (푈), then (푓푠)|푈 (푥) = 푓(푥)푠(푥)|푈 determines a multiplication of a section 푠 by an element 푓 ∈ 풪푋 (푈). For any open sets 푈, 푉 ⊂ 푋 with 푉 ⊂ 푈, the restriction maps 휌푈푉 : ℒ퐸(푈) → ℒ퐸(푉 ) are given by restricting the sections from 푈 to 푉 . Moreover, note that the restriction homomorphisms are compatible with the module structure. Indeed, if 푓 ∈ 풪푋 (푈) and 푠 ∈ ℒ퐸(푈) then 푓|푉 ·푠|푉 = 푓푠|푉 . Now, we will see that ℒ퐸 is a locally free sheaf. Since 퐸 is a vector bundle, let 푈푖 be open −1 푟 subsets of 푋 so that 휑푈푖 : 휋 (푈푖) → 푈푖 × A is an isomorphism. If 푠 is a section of 푈푖 we 푟 obtain a map 푈푖 → A as the composition 휑 −1 푈푖 푟 휋2 푟 푈푖 → 휋 (푈푖) −−→ 푈푖 × A −→ A 푟 ∼ 1 1 where 휋2 is the projection onto the second coordinate. Since A = A × ... × A , a section of 푈푖 1 is an 푟-tuple of maps to A . Therefore, ℒ퐸 is a locally free sheaf of rank 푟.
∨ Let ℱ be a locally free sheaf of finite rank, define the dual sheaf of ℱ by ℱ := 퐻표푚푂푋 (ℱ, 푂푋 )
Theorem 1.1.16. Let ℱ be a locally free sheaf of 풪푋 -modules of finite rank. Then for any 풪푋 -module 풢 we have ∼ ∨ Hom푂푋 (ℱ, 풢) = ℱ ⊗풪푋 풢
Proof. By the universal property of sheafification we need to find an isomorphism fromthe ∨ presheaf of ℱ ⊗풪푋 풢 to Hom푂푋 (ℱ, 풢) in order to show this isomorphism. Define a morphism 푝푟푒 휑 : Hom(ℱ, 풪푋 ) ⊗ 풢 → Hom(ℱ, 풢) CHAPTER 1. PRELIMINARIES 17 for any open set 푈 ⊂ 푋 as follows
푝푟푒 휑푈 : Hom풪푋|푈 (ℱ|푈 , 풪푋|푈 ) ⊗ 풢(푈) → Hom풪푋|푈 (ℱ|푈 , 풢|푈 ) (푡) 휓 ⊗ 푡 → (휓 : ℱ|푈 → 풢|푈 ) where 휓 and 휓(푡) are morphisms of sheaves, and for any 푉 ⊂ 푈 in 푋, 휓(푡) is defined by
(푡) 휓푉 : ℱ(푉 ) → 풢(푉 )
푠 → 휓푉 (푠)(푡|푉 )
We have that the collection {휑푈 } is an isomorphism if and only if the induced maps on the stalks are an isomorphism. Since ℱ is locally free, the stalks are free, that is, they are isomorphic to 풪푥,푋 . Then, the morphisms are exactly the canonical isomorphisms of 풪푥,푋 modules. Therefore, the induced morphisms are isomorphisms.
We are ready to establish a correspondence between vector bundles and locally free sheaves. The next theorem can be found in [16, p. 58]. Theorem 1.1.17. Let 퐸 be a vector bundle of rank 푟 over a variety 푌 . The correspondence
퐸 → ℒ퐸 establishes a one-to-one correspondence between vector bundles of rank 푟 and locally free sheaves of rank 푟 up to isomorphism. We can also deduce from this theorem that line bundles correspond to invertible sheaves.
Notation. Usually we denote by ℰ the sheaf corresponding to a vector bundle 퐸. We will use any of these notations indistinctly.
Proposition 1.1.18. Let ℱ be a locally free 풪푋 -module of rank 1. Then, ∼ Hom풪푋 (ℱ, ℱ) = 풪푋
Proof. Let 휑 ∈ Hom푂푋 (ℱ, ℱ)(풰) for some open set 풰 ⊂ 푋 small enough. We can see that 휑(푓) = 푓휑(1) where 1 is the identity element of 풪푋 |푈 , and then 휑 is determined by 휑(1). Therefore, there are ℱ|푈 possible elements all resulting in a homomorphism. However, since ℱ ∼ ∼ ∼ is a locally free sheaf of rank 1, we have that ℱ|푈 = 풪푋 |푈 , and so Hom풪푋 (ℱ, ℱ)(풰) = ℱ|푈 = 풪푋 |푈 .
Proposition 1.1.19. Let ℱ, and 풢 be invertible sheaves on an algebraic variety 푋, then ℱ ⊗풢 is also an invertible sheaf. If ℱ is an invertible sheaf on 푋, then there exists an invertible sheaf −1 −1 ∼ ℱ on 푋 such that ℱ ⊗ ℱ = 풪푋 . ∼ ∼ Proof. We have that ℱ, and 풢 are invertible sheaves, then ℱ ⊗풢 = 풪푋 ⊗풪푋 = 풪푋 , and so the first statement follows. For the second statement, let ℱ be any invertible sheaf. Take ℱ −1 to be ∨ ∨ ∼ the dual sheaf ℱ = Hom(ℱ, 풪푋 ), then by Theorem 1.1.16 we have that ℱ ⊗ℱ = Hom(ℱ, ℱ). ∼ Also, by Proposition 1.1.18 Hom(ℱ, ℱ) = 풪푋 , and so the result follows. CHAPTER 1. PRELIMINARIES 18
Definition 1.1.20. Let 푋 be an algebraic variety, we define the Picard group of 푋, Pic(푋), to be the group of isomorphism classes of invertible sheaves on 푋, with the operation of tensor product ⊗.
The Proposition 1.1.19 shows that in fact Pic(푋) is a group. The identity element is 풪푋 . ∼ ∼ ∨ ∼ Furthermore, if ℱ ⊗ 풢 = 풪푋 = ℱ ⊗ ℋ, then tensoring both sides by ℱ gives us 풢 = ℋ. So, the inverse element of ℱ is unique.
1.2 Divisors
The notion of divisors is important for the understanding of the intrinsic geometry of a variety. This notion was introduced by Dedekind and Weber, see [6, p. 828]. In this section we will define Weil divisors and Cartier divisors, and we will explain the connection between Weil divisors, Cartier divisors and invertible sheaves.
1.2.1 Weil Divisors We will use subvarieties of codimension one to store information of a variety. We can obtain information about the zeroes and poles of a function defined over a variety. We will start by defining Weil divisors and we will give some basic properties of them. After that, wewillshow how we can associate a Weil divisor with a function. Definition 1.2.1 (Weil divisors). Let 푋 be a variety. A Weil divisor is a finite formal sum of codimension one subvarieties of ∑︀푘 푋. This is denoted by 퐷 = 푖=1 푎푖푌푖 where each of the 푌푖 are subvarieties of 푋 of codimension one. Weil divisors can be added as follows. 푘 푘 푘 ∑︁ ∑︁ ∑︁ 푛푖푌푖 + 푚푖푌푖 = (푛푖 + 푚푖)푌푖 푖=1 푖=1 푖=1 This forms a free abelian group Div(푋) of codimension one subvarieties of 푋 defined as Div(푋) = {퐷|퐷 is a Weil divisor } . An irreducible codimension 1 subvariety 푌푖 taking with multiplicity 1 is called a prime divisor. The support of a divisor is the union of all 푌푖 with 푎푖 ≠ 0.
Definition 1.2.2. The degree deg 퐷 of a divisor 퐷 = 푎1푌1 + ... + 푎푚푌푚 is defined to be the integer 푎1 + ... + 푎푚. The degree function deg : Div(퐶) → Z is a group homomorphism.
Definition 1.2.3. A divisor 퐷 is effective(denoted 퐷 ≥ 0) if every term in the divisor is non-negative. That is 푎푖 ≥ 0 for all 푖. CHAPTER 1. PRELIMINARIES 19
One can induce a partial order on divisors by setting 퐷1 ≥ 퐷2 ⇐⇒ 퐷1 − 퐷2 ≥ 0.
Example 1.2.4. Let 푋 be the zero set of 푓, where 푓 is defined by 푓(푥, 푦, 푧) = 푥2푦 − 푧3. A codimension one subvariety of 푋 is a closed point. Then, 퐷 := 4[1 : 1 : 1] − [0 : 1 : 0] is a divisor on 푋. The support of this divisor is {[1 : 1 : 1], [0 : 1 : 0]}, and deg(퐷) = 3. Also, this divisor is not effective since the coefficient of [0 : 1 : 0] is negative. Now, we want to define the divisor of a rational function. For this, we need to define avaluation of a field 푘. Definition 1.2.5. Let 푘 be a field and let 퐺 be a totally ordered abelian group. A valuation of 푘 with values in 퐺 is a map 푣 : 푘 ∖ {0} → 퐺 such that (i) 푣(푥푦) = 푣(푥) + 푣(푦). (ii) 푣(푥 + 푦) ≥ min(푣(푥), 푣(푦)). for all 푥, 푦 ∈ 푘 ∖ {0}. If 푣 is a valuation, the set 푅 = {푥 ∈ 푘|푣(푥) ≥ 0} ∪ {0} will be called the valuation ring of 푣. A valuation 푣 is discrete if its value group 퐺 is the integers. From now and on we will consider discrete valuations.
Proposition 1.2.6. Let 푘 be a field, and let 푣 : 푘 − {0} → Z be a discrete valuation. If 푥, 푦 ∈ 푘 ∖ {0} with 푣(푥) ≠ 푣(푦) then 푣(푥 + 푦) = 푚푖푛{푣(푥), 푣(푦)}.
Proof. Since 푣(푥) ≠ 푣(푦), consider 푣(푥) < 푣(푦). Suppose 푣(푥 + 푦) ≠ 푚푖푛{푣(푥), 푣(푦)}. We have that
푣(푥 + 푦) ≥ 푚푖푛{푣(푥), 푣(푦)} = 푣(푥).
For every 푎 ∈ 푘 we have 푣(푎푦) = 푣(푦). Therefore, 푣(−푦) = 푣(푦). Then, 푣(푥) = 푣((푥 + 푦) − 푦) ≥ 푚푖푛{푣(푥 + 푦), 푣(푦)} > 푣(푥). which is absurd. So, 푣(푥 + 푦) = 푚푖푛{푣(푥), 푣(푦)}.
Notation. We will denote by 푘(푋) the function field of 푋. The elements of 푘(푋) are called rational functions on 푋. Also, the set of non-zero elements of 푘(푋) is going to be denoted by 푘(푋)*.
Let 푋 be a smooth algebraic variety and 푌 a subvariety of 푋 of codimension 1. Then, the local ring 풪푌,푋 of 푋 along 푌 is a discrete valuation ring. The valuation on this ring is defined 푛 as follows: Let 푓 ∈ 풪푌,푋 , if 푀 = (푡) is the maximal ideal for 풪푌,푋 and 푓 = 푢푡 for some unit * 푢 ∈ 풪푌,푋 , then the valuation of 풪푌,푋 is 푣푌 (푓) := 푛. If 푣푌 (푓) is positive, we say 푓 has a zero along 푌 , of that order; if it is negative, we say 푓 has a pole along 푌 of order −푣푌 (푓). Since there are only finitely many 푌 with 푣푌 (푓) ≠ 0 we have the following definition. CHAPTER 1. PRELIMINARIES 20
Definition 1.2.7. Let 푋 be a variety, the divisor of 푓 ∈ 푘(푋)* , denoted by (푓) or 푑푖푣(푓) is defined as the sum ∑︁ 푣푌 (푓)푌, where the sum takes place over all the irreducible codimension 1 subvarieties 푌 for which 푣푌 (푓) ≠ 0. A divisor of the form 퐷 = (푓) for some 푓 ∈ 푘(푋)* is called a principal divisor. The subset of 퐷푖푣푋 of all divisors of the form (푓) is a subgroup of 퐷푖푣푋. If 푓, 푔 ∈ 푘(푋)*, then (푓/푔) = (푓) − (푔).
Definition 1.2.8. Two divisors 퐷1 and 퐷2 are said to be linearly equivalent if 퐷1 = 퐷2+(푓) for some 푓 ∈ 푘(푋)*. Definition 1.2.9. The divisor class group 퐶푙(푋) of 푋 is defined to be the group Div(푋) modulo linear equivalence . Example 1.2.10. Let 푓(푥, 푦, 푧) = 푦2푧 − (푥 − 3푧)(푥 − 4푧) and suppose 푋 = 푍(푓). Let 퐷1 := [3 : 0 : 1] − [0 : 1 : 0] and 퐷2 := [0 : 1 : 0] − [4 : 0 : 1]. We have that 퐷1 is linearly equivalent to 퐷2 since
(푦) = [3 : 0 : 1] + [4 : 0 : 1] − 2[0 : 1 : 0] = 퐷1 − 퐷2.
Definition 1.2.11 (Divisor on a Curve). ∑︀ Let 푋 be a nonsingular projective curve. A divisor on 푋 is a linear combination 퐷 = 푘푖푥푖 of points 푥푖 with coefficients 푘푖 ∈ Z.
1.2.2 Cartier Divisors Cartier divisors are a different way of store information of a variety. Instead of considering codimension 1 subvarieties, we consider (locally) the functions that respectively define those subvarieties. We will also see that there is a one-to-one correspondence between Weil divisors and Cartier divisors when 푋 is a smooth variety. Definition 1.2.12 (Cartier divisors). Let 푋 be a variety and 퐼 an indexing set. A Cartier divisor is an equivalence class of collections of pairs [{(푈푖, 푓푖)}푖∈퐼 ] such that ⋃︀ (i) 푖∈퐼 푈푖 = 푋 with each 푈푖 open. * (ii) 푓푖 ∈ 푘(푋) .
(iii) 푓푖/푓푗 and 푓푗/푓푖 are both regular on 푈푖 ∩ 푈푗.
(iv) The equivalence relation ∼ is defined by {(푈푖, 푓푖)}푖∈퐼 ∼ {(푉푗, 푔푗)}푗∈퐽 ⇔ 푓푖/푔푗 is regular on 푈푖 ∩ 푉푗 for every 푖 ∈ 퐼 and 푗 ∈ 퐽. Cartier divisors form a group denoted CaDiv(푋) by defining the sum of two divisors as
{(푈푖, 푓푖)}푖∈퐼 + {(푉푗, 푔푗)}푗∈퐽 = {(푈푖 ∩ 푉푗, 푓푖푔푗)}푖∈퐼,푗∈퐽 CHAPTER 1. PRELIMINARIES 21
The notion of a Cartier divisor can be reinterpret in the language of sheaves. Let 풦푋 be the sheaf of rational functions, and 풪푋 be the sheaf of regular functions, then we have the next definition. Definition 1.2.13. Let 푋 be a variety, a Cartier divisor on 푋 is a global section of the * * quotient sheaf 풦푋 /풪푋 , where the * denotes the set of invertible elements of their respective sheaves.
1.2.3 Correspondence between Weil Divisors and Cartier Divisors We are ready to state the following theorem that establishes a correspondence between Cartier divisors and Weil divisors. This theorem can be found in [8, p. 141]. Theorem 1.2.14. Let 푋 be a smooth variety. Then there exists an isomorphism 휑 between the group of Cartier divisors and the group of Weil divisors.
휑 : CaDiv(푋) → Div(푋).
1.2.4 The invertible sheaf associated to a divisor
Definition 1.2.15. Let 퐷 := {(풰푖, 푓푖)}푖∈퐼 be a Cartier divisor. We define a subsheaf ℒ(퐷) of the sheaf of rational functions 풦푋 by taking ℒ(퐷) to be the sub-풪푋 -module of 풦푋 generated by −1 푓푖 on 푈푖. That is, 1 ℒ(퐷)(풰푖) := 풪푋 (풰푖) 푓푖
={푓 ∈ 풦푋 (풰푖)|퐷 + 푑푖푣(푓) ≥ 0}
The sheaf ℒ(퐷) is called the invertible sheaf associated to a divisor 퐷 and is also denoted by 풪푋 (퐷). ∼ Theorem 1.2.16. Let 퐷1, and 퐷2 be Cartier divisors on a variety 푋. Then ℒ(퐷1 − 퐷2) = ∨ ℒ(퐷1) ⊗ ℒ(퐷2) .
−1 Proof. Let 퐷1 = {(풰푖, 푓푖)}푖∈퐼 and 퐷2 = {(풱푗, 푓푗)}푗∈퐽 . Then, 퐷1 −퐷2 = {(풰푖 ∩풱푗, 푓푖푔푗 )}푖∈퐼,푗∈퐽 . ∼ It is enough to show that ℒ(퐷1 − 퐷2) = ℒ(퐷1) ⊗ ℒ(−퐷2). By the universal property of 푝푟푒 sheafification we need to construct an isomorphism from ℒ(퐷1) ⊗ ℒ(−퐷2) to ℒ(퐷1 − 퐷2), 푝푟푒 where ℒ(퐷1) ⊗ ℒ(−퐷2) denotes the presheaf of ℒ(퐷1) ⊗ ℒ(−퐷2). For any open set 풰 ⊂ 푋, define a ring homomorphism 휑풰 by
푝푟푒 (ℒ(퐷1) ⊗ ℒ(−퐷2))(풰 ∩ 풰푖 ∩ 풱푗) → ℒ(퐷1 − 퐷2)(풰 ∩ 풰푖 ∩ 풱푗) 푓 ⊗ 푔 → 푓푔
1 1 1 The function 푓푔 is in ℒ(퐷1−퐷2)(풰 ∩풰푖∩풱푗) since 푓 = ℎ1 and 푔 = ℎ2 and so 푓푔 = ℎ1ℎ2 ∈ 푓푖 푔푗 푓푖푔푗 ℒ(퐷1 −퐷2)(풰 ∩풰푖 ∩풱푗). The maps are well defined on the overlaps by the properties of Cartier divisors. Furthermore, we can see that this map is an isomorphism of rings, then {휑풰 } is an ∼ ∨ ∼ isomorphism of sheaves. So, ℒ(퐷1) ⊗ ℒ(−퐷2) = ℒ(퐷1) ⊗ ℒ(퐷2) = ℒ(퐷1 − 퐷2). ∼ Corollary 1.2.17. Let 푋 be a variety, we have that ℒ(0) = 풪푋 . CHAPTER 1. PRELIMINARIES 22
Proof. Let 퐷 be a Cartier divisor on 푋. Then, by Theorem 1.2.16 we get that ℒ(0) = ℒ(퐷 − 퐷) ∼= ℒ(퐷)∨ ⊗ ℒ(퐷). Also, since ℒ(퐷) is an invertible sheaf we can apply Proposition 1.1.18 ∨ ∼ ∼ ∼ to obtain that ℒ(퐷) ⊗ ℒ(퐷) = Hom풪푋 (ℒ(퐷), ℒ(퐷)) = 풪푋 . Therefore, ℒ(0) = 풪푋 .
There is a correspondece between Cartier divisors and invertible sheaves that is stated in the following theorem. Theorem 1.2.18. For any Cartier divisor 퐷, we have that ℒ(퐷) is an invertible sheaf. The map 퐷 → ℒ(퐷) gives a one to one correspondence between Cartier divisors on 푋 and invertible subsheaves on 풦푋 .
−1 Proof. Notice that the map 풪푈푖 → ℒ(퐷)|푈 푖, defined by 1 → 푓푖 is an isomorphism. Therefore, ℒ(퐷) is an invertible sheaf. Now, we will show that the Cartier divisor 퐷 can be recovered from ℒ(퐷) as follows. Start with an invertible subsheaf of 풦푋 , say ℱ. Take 푓푖 on 풰푖 to be the inverse of a local generator of the sheaf on 풰푖. Doing this for an open cover will give us a Cartier divisor, say 퐷. Applying 1 the map gives us ℒ(퐷) = 풪푋 (풰푖) = ℱ(풰푖) by choice of 푓푖 as the inverse of a local generator, 푓푖 so we have a 1 − 1 correspondence.
To summarize, from Theorem 1.1.17 and Theorem 1.2.18 we have the following three equivalent notions(up to isomorphism): line bundles ⇔ locally free sheaves of rank one ⇔ divisors
Definition 1.2.19. The degree of a line bundle is the degree of the divisor associated, and the degree of a vector bundle 퐸 of rank 푟 > 1 is defined by deg(퐸) = deg(det(퐸)), where det(퐸) is the determinant line bundle of 퐸. The following theorem states that starting with line bundles, and by successively taking extensions we can obtain all vector bundles. The proof of this theorem will be done in the next chapter because it needs more concepts. Theorem 1.2.20. Let 퐿 be a line bundle over 푋, 퐸 a vector bundle of rank 푟 over 푋, and 퐹 a vector bundle of rank 푟 − 1 over 푋. Then, we have an exact sequence 0 → 퐿 → 퐸 → 퐹 → 0
Whenever we have an exact sequence as in the theorem above we will also have that deg(퐸) = deg(퐿) + deg(퐹 ). We will prove this in Chapter 2, Proposition 2.2.16.
The next lemma will serve us to prove the Riemann–Roch Theorem in the next chapter.
Lemma 1.2.21. Let 푋 be a topological space, 푃 ∈ 푋 a point, 푈 ⊂ 푋 an open subset. Let 풦(푃 ) be the skyscraper sheaf defined as ⎧ ⎨푘 if 푃 ∈ 푈 풦(푃 )(푈) = ⎩0 if 푃∈ / 푈 CHAPTER 1. PRELIMINARIES 23
Therefore, we have an exact sequence
휓 휑 0 → ℒ(퐷) −→ℒ(퐷 + 푃 ) −→풦(푃 ) → 0
Proof. Define 휓 as the natural injection from ℒ(퐷) to ℒ(퐷 + 푃 ) . Define 휑 from ℒ(퐷 + 푃 ) to 풦(푃 ) as follows ⎧ ⎨0 if 푃∈ / 풰 휑|풰 (푓) = ⎩푓(푃 ) if 푃 ∈ 풰
So this map is onto, and we get an exact sequence
휓 휑 0 → ℒ(퐷) −→ℒ(퐷 + 푃 ) −→풦(푃 ) → 0.
Next, we will define the linear system L(퐷). This space is very important because the Riemman-Roch Theorem computes its dimension. Definition 1.2.22. Let 푋 be an algebraic variety and let 퐷 ∈ Div(푋) be a divisor. We can associate to 퐷 a vector space L(퐷) over 푘 by
L(퐷) = {푓 ∈ 푘(푋)*|퐷 + 푑푖푣(푓) ≥ 0} ∪ {0}.
We denote by 푙(퐷) the dimension of L(퐷). ∑︀푟 ∑︀푠 This definition can be interpreted as: if 퐷 = 푖=1 푛푖푃푖 − 푗=1 푚푗푄푗 with 푛푖 > 0 and 푚푗 > 0 then L(퐷) consists of all elements 푓 ∈ 푘(푋) such that
• 푓 has zeros of order ≥ 푚푗 at 푄푗, for 푗 = 1, ..., 푠, and
• 푓 may have poles only at 푃1, ..., 푃푟 with pole order at 푃푖 being bounded by 푛푖, 푖 = 1, ..., 푟.
Note that 푓 ∈ L(퐷) if and only if 푣푃 (푓) ≥ −푣푃 (퐷) for all 푃 ∈ 푋. The space L(푚푃 ) is the set of functions 푓 ∈ 푘(푋) having only poles at 푃 and with pole order being bounded by 푚. Proposition 1.2.23. Let 푋 be an algebraic variety, 퐷 a Weil divisor on 푋, and denote by Γ(푋, ℒ(퐷)) the set of global sections of the invertible sheaf ℒ(퐷). We have the next isomorphism
Γ(푋, ℒ(퐷)) ∼= L(퐷).
∑︀ Proof. Let 퐷 = 푎푖푌푖 a Weil divisor on 푋, and let {(푓푖, 푈푖)}푖∈퐼 be the Cartier divisor associated to 퐷. L Let 푔 ∈ (퐷). From 푣푌푖 (푔) ≥ −푎푖 we have that 푔푖 = 푔푓푖 ∈ 풪푋 (풰푖). This functions satisfy −1 푔푖 = 푓푖푓푗 푔푗, then the functions 푔푖 define a global section 푔푓 ∈ Γ(푋, ℒ퐷). −1 Reciprocally, if 푠 ∈ Γ(푋, ℒ(퐷)) we obtain 푠푖 ∈ 풪(풰푖) satisfying 푠푖 = 푓푖푓푗 푠푗. Therefore, we * can define 푔 ∈ 푘(푋) by 푔|풰푖 = 푠푖/푓푖. Since the 푓푖’s define 퐷, we have that 푔 is rational in 푋 ∖퐷 L and 푣푌푖 (푔) = 푣푌푖 (푠푖) − 푣푌푖 (푓푖) ≥ 푣푌푖 (푓푖) = −푎푖. That is, 푔 ∈ (퐷).
The next theorem can be found in [18, p. 18]. CHAPTER 1. PRELIMINARIES 24
Theorem 1.2.24. Let 푋 be a non-singular curve and let 푓 ∈ 푘(푋). Then
푑푒푔(푑푖푣(푓)0) = 푑푒푔(푑푖푣(푓)∞) = [푘(푋): 푘(푓)].
Lemma 1.2.25. Let 푋 be a nonsingular curve, and let 퐷 be a divisor on 푋. Then (i) L(0) = 푘. (ii) If 퐷 < 0 then L(퐷) = {0}.
Proof. (i) We have that 푘 ⊆ L(0) because 푑푖푣(푥) = 0 for 0 ≠ 푥 ∈ 푘. Conversely, if 0 ≠ 푓 ∈ L(0) then 푑푖푣(푓) ≥ 0. This is, 푓 has no pole, so 푓 ∈ 푘. (ii) Assume that there exists an element 0 ≠ 푓 ∈ L(퐷). Then 푑푖푣(푓) ≥ −퐷 > 0, which implies that 푓 has at least one zero but no pole. This is impossible.
Lemma 1.2.26. Let 푋 be a nonsingular curve, and let 퐷 be a divisor on 푋. Then
ℓ(퐷) − ℓ(퐷 − 푃 ) ≤ 1.
Proof. Let be 푛 = 푣푃 (퐷) and 푡 a local parameter of 푃 . Consider the map
휑 : L(퐷) → 푘 푓 → (푡푛푓)(푃 )
Note that 휑 is a homomorphism whose ker(휑) are all the elements 푔 ∈ L(퐷) such that 푛 푛 푛 푛 푛 (푡 푔)(푃 ) = 0, i.e. 푃 is a zero of 푡 푔, then 푣푃 (푡 푔) > 0. Since 푣푃 (푡 푔) = 푣푃 (푡 ) + 푣푃 (푔) = 푛 + 푣푃 (푔) = 푣푃 (퐷) + 푣푃 (푔) ≥ 1 we have that ker(휑) = L(퐷 − 푃 ). Therefore, L(퐷)/L(퐷 − 푃 ) ∼= 퐼푚(휑) ⊆ 푘. Consequently, the dimension of the quotient L(퐷)/L(퐷 − 푃 ) is at most equal to one, i.e. 푙(퐷) − 푙(퐷 − 푃 ) ≤ 1.
Proposition 1.2.27. Let 퐷 be a divisor on a nonsingular curve 푋. The vector space L(퐷) is finite dimensional.
Proof. The proof reduces to the case 퐷 ≥ 0. In fact, let 퐷 = 퐷1 − 퐷2 with 퐷1, 퐷2 ≥ 0. We ′ will see that L(퐷) ⊂ L(퐷1). Take 푓 ∈ L(퐷), then 푑푖푣(푓) + 퐷1 − 퐷2 = 퐷 ≥ 0. Therefore, ′ 푑푖푣(푓) + 퐷1 = 퐷 + 퐷2 ≥ 0, that is, 푓 ∈ L(퐷1). Now, if 푑 = deg(퐷) ≥ 0 we have that 푙(퐷) ≤ 푙(퐷 − (푑 − 1)푃 ) + 푑 − 1 by Lemma 1.2.26. Since deg(퐷 − (푑 − 1)푃 ) < 0 by part (ii) of Lemma 1.2.25 we get 푙(퐷 − (푑 − 1)푃 ) = 0. Thus, 푙(퐷) ≤ 푑 − 1 and the vector space L(퐷) is finite dimensional.
1.2.5 Canonical Class In this subsection, let 퐴 be a commutative ring with identity, 퐵 an 퐴-algebra, and 푀 be a 퐵-module. Definition 1.2.28. An 퐴-derivation of 퐵 into 푀 is a map 푑 : 퐵 → 푀 such that (i) 푑(푎) = 0 CHAPTER 1. PRELIMINARIES 25
(ii) 푑(푏 · 푏′) = 푏 · 푑푏′ + 푏′ · 푑푏 (iii) 푑(푏 + 푏′) = 푑푏 + 푑푏′ for all 푎 ∈ 퐴 and 푏, 푏′ ∈ 퐵.
Definition 1.2.29. The module of relative forms of 퐵 over 퐴 is a 퐵-module Ω퐵/퐴, together with an 퐴-derivation 푑 : 퐵 → Ω퐵/퐴, which satisfies the following universal property: for any 퐵-module 푀, and for any 퐴-derivation 푑′ : 퐵 → 푀, there exists a unique 퐵-module ′ homomorphism 푓 :Ω퐵/퐴 → 푀 such that 푑 = 푓 ∘ 푑.
푑 퐵 Ω퐵/퐴
푓 푑′ 푀
Now, we want to describe the sheaf of differentials. Definition 1.2.30. Let 푋 be a topological space. Let ℱ and 풢 be sheaves of rings with 푓 : 푝푟푒 ℱ → 풢 a morphism of sheaves. Consider 풢 as an ℱ-module. Define the presheaf Ω풢/ℱ by
풰 → Ω풢(푈)/ℱ(푈) with the usual restrictions maps. This map is a ℱ(풰)-derivation and so it factors through 푝푟푒 푝푟푒 Ω풢/ℱ (풰) giving the restriction maps for Ω풢/ℱ . Now, we sheafify it to get Ω풢/ℱ the sheaf of relative differentials. Definition 1.2.31. Let 푓 : 푋 → 푌 be a morphism of varieties. We can retrieve a morphism −1 −1 푓 풪푌 → 풪푋 using the inverse image sheaf, which turns 풪푋 into a 푓 풪푌 module. Define
−1 Ω푋/푌 to be Ω풪푋 /푓 풪푌 . The following theorem can be found in [8, p. 177].
Theorem 1.2.32. Let 푋 be a smooth algebraic variety over 푘. Then Ω푋/푘 is a locally free sheaf of rank 푛 = dim 푋. Definition 1.2.33. Let 푋 be a smooth algebraic variety of dimension 푛. Define the Canonical ⋀︀푛 sheaf of 푋 to be 휔푋 = Ω푋/푘, the 푛th exterior power of the sheaf of differentials Ω푋/푘.
By Theorem 1.2.32 the canonical sheaf 휔푋 is an invertible sheaf, so by Theorem 1.2.18 there is an associated divisor to 휔푋 . This divisor will be called a canonical divisor and denoted 퐾푋 . A canonical divisor is unique up to linear equivalence. Definition 1.2.34. Let 푋 be a smooth projective variety over 푘, we define the geometric genus of 푋 to be 푔 := dim푘 Γ(푋, 휔푋 ), where Γ(푋, 휔푋 ) denotes the set of global sections of the canonical sheaf. Next, we will give the definition of local parameters which will serve us to give examples in Chapter 3. Definition 1.2.35. Let 푥 be a smooth point on an algebraic variety 푋 with dim(푋) = 푛. Then, 푡1, ..., 푡푛 ∈ 풪푥,푋 are called local parameters at 푥 if the 푡푖 ∈ 푚푥 := {푓 ∈ 푘(푋)|푓(푥) = 0} , 2 and if they give a basis of 푚푥/푚푥. CHAPTER 1. PRELIMINARIES 26
1.3 Stable Bundles
Definition 1.3.1. The slope of a vector bundle 퐸 is the rational number 휇(퐸) = deg(퐸)/rk(퐸) and is denoted by 휇(퐸). Definition 1.3.2. A vector bundle 퐸 on 푋 is stable if for every proper non-zero subbundles 퐹 of 퐸 we have that 휇(퐹 ) < 휇(퐸). Also, a vector bundle 퐸 on 푋 is semistable if for every proper non-zero subbundles 퐹 of 퐸 we have the inequality 휇(퐹 ) ≤ 휇(퐸).
Example 1.3.3. Line bundles are stable because they do not have any proper subbundles. If there is a non-trivial extension
0 → 퐿0 → 퐸 → 퐿1 → 0 where 퐿0,퐿1 are line bundles of degree 0 and 1 respectively, then 퐸 is stable.
Proof. In fact, since the degree is additive in exact sequences, we have
deg(퐸) = deg(퐿0) + deg(퐿1) = 1.
deg(퐸) 1 Then, 휇(퐸) = rk(퐸) = 2 . Let 푀 ⊂ 퐸 be an arbitrary subsheaf. If rk(푀) = 2 then 퐸/푀 is a sheaf of dimension zero of length, say 푙 > 0 and
휇(푀) = 휇(퐸) − 푙/2 < 휇(퐸).
If rk(푀) = 1 consider the composition 푀 → 퐿1. This is either zero or injective. If it is zero, 푀 ⊂ 퐿0 and therefore 1 휇(푀) ≤ 휇(퐿0) = 0 < = 휇(퐸). 2
If 푀 ⊂ 퐿1, then 휇(푀) ≤ 휇(퐿1) = 1. If 휇(푀) = 1, then 푀 = 퐿1 and 푀 would provide a splitting of the extension which contradicts the assumption. Therefore,
휇(푀) ≤ 0 < 1/2 = 휇(퐸). 27
Chapter 2
Riemann–Roch Theorem
In this chapter we start by defining the 푛th Čech cohomology group of a topological space 푋. Next, we will state some basic results of cohomology that will serve us to prove the Riemmann–Roch Theorem. This theorem was a response to the well-known Riemann–Roch problem which is to compute the dimension of the space L(퐷) on an algebraic curve 푋. Riemann stated that:
푙(퐷) ≥ deg(퐷) + 1 − 푔 and Roch provided the error term
푙(퐷) − 푙(퐾푋 − 퐷) = deg(퐷) + 1 − 푔 giving origin to the Riemann–Roch Theorem which was first called like that by Brill and Noether [2].
2.1 Čech Cohomology
Definition 2.1.1. Let 푋 be a topological space, ℱ a sheaf of abelian groups on 푋, and let U = {푈푖}{푖∈퐼} be an open cover of 푋. Assume that 퐼 is an ordered set. For all 푝 ≥ 0 we define the Abelian group 푝 ∏︁ 풞 (ℱ) = ℱ(푈푖0 ∩ · · · ∩ 푈푖푝 ) 푖0<···<푖푝
푝 An element 푓 ∈ 풞 (ℱ) is a collection 푓 = (푓푖0,...,푖푝 ) of sections of ℱ(푈푖0 ∩ · · · ∩ 푈푖푝 ).The addition on this group is defined componentwise. Definition 2.1.2. For every 푝 ≥ 0, the boundary operators are defined as
푑푝 : 풞푝(ℱ) → 풞푝+1(ℱ) 푝+1 푓 → 푑푝푓 ∑︁ − 푘푓 | 푖0,...,푖푘−1,푖푘+1,...,푖푝+1 ( )푖0,...,푖푝+1 = ( 1) 푖0,...,푖푘−1,푖푘+1,...,푖푝+1 푈푖0 ∩...∩푈푖푝+1 푘=0 CHAPTER 2. RIEMANN–ROCH THEOREM 28
The maps 푑푝 are called boundary operators since 푑푝+1 ∘ 푑푝 = 0 for every 푝 ≥ 0. Therefore, we have defined a sequence of Abelian groups and homomoprhisms, usually called a complex of Abelian groups.
푑0 푑1 푑2 풞0(ℱ) −→풞1(ℱ) −→풞2(ℱ) −→· · · 0 ∏︀ 0 For example, let 풞 = 훼∈퐼 ℱ(푈훼) so that an element of 풞 has the form {푓훼} with 푓훼 ∈ ℱ(푈훼). 1 ∏︀ Also, 풞 = 훼,훽∈퐼 ℱ(푈훼 ∩ 푈훽) where an element looks like {푓훼,훽}. Similarly, 2 ∏︀ 풞 = 훼,훽,훾∈퐼 ℱ(푈훼 ∩ 푈훽 ∩ 푈훾) and so on. The first two differentials are
0 0 1 • 푑 : 풞 → 풞 given by {푓훼} → {푓훼|푈훼∩푈훽 −푓훽|푈훼∩푈훽 ∈ ℱ(푈훼 ∩ 푈훽)} 1 1 2 • 푑 : 풞 → 풞 given by {푓훼,훽} → {푓훽,훾 − 푓훼,훾 + 푓훼,훽 ∈ ℱ(푈훼 ∩ 푈훽 ∩ 푈훾)} Definition 2.1.3. Let U be an open cover of a topological space 푋, and ℱ be a sheaf of abelian groups on 푋. Consider the homomorphisms 푑푝 : 풞푝(ℱ) → 풞푝+1(ℱ) . The 푝th Čech cohomology group of ℱ with respect to an open cover U is defined as 퐻푝(U, ℱ) := ker 푑푝/im 푑푝−1 where 풞푝(ℱ) and 푑푝 are zero for 푝 < 0. Since 푑푝+1 ∘ 푑푝 = 0, we can prove that im 푑푝+1 ⊂ ker 푑푝. Then, the above definition is well-defined.
Example 2.1.4. We have 퐻0(U, ℱ) = Γ(푋, ℱ) = 퐻0(푋, ℱ). Indeed, by definition 0 0 −1 ∼ 퐻 (U, ℱ) := ker 푑 /im 푑 = {(푓푖): 푓푖 − 푓푗|푈푖∩푈푗 = 0}
By the sheaf condition, the 푓푖 glue together to give a global section 푓 ∈ Γ(푋, ℱ) such that 0 푓푖 = 푓|푈푖 . Reciprocally, if 푓 ∈ Γ(푋, ℱ) , then we have (푓푖) ∈ ker 푑 where 푓푖 = 푓|푈푖 . In conclusion, 퐻0(U, ℱ) = Γ(푋, ℱ) = 퐻0(푋, ℱ). Now, we will extend this notion to one that is independent of the chosen open cover.
Definition 2.1.5. Let U = {푈푖}푖∈퐼 , and V = {푉휆}휆∈Λ be open coverings of 푋. We said that V is a refinement of U if for every 푉휆 there exists 푈훼(휆) ∈ U such that 푉휆 ⊂ 푈훼(휆) where 훼 :Λ → 퐼. We denote it by 풱 > 풰. Definition 2.1.6. 푝 푝 푝 For 훽 : 퐼 → Λ satisfying 푉훽푖 ⊂ 푈푖, define a homomorphism 휏훽 : 풞 (U, ℱ) → 풞 (V, ℱ) by 푝 휏 ((푓푖 ,...,푖 )) = (푓훽(푖 ),...,훽(푖 )|푉 ∩...∩푉 ) 훽 0 푝 0 푝 훽(푖0) 훽(푖푝)
푝 푝 푝 푝 푝 푝 푝+1 푝 푝 푝 If 푑풰 is 푑 for 풞 (U, ℱ), and 푑풱 is 푑 for 풞 (V, ℱ), then 휏훽 ∘푑U = 푑V ∘휏훽 . Therefore, the maps 푝 푛 {휏훽 } induce an additive group homomorphism on the cohomology groups going from 퐻 (U, ℱ) 푛 ᨀ 푛 푛 to 퐻 (V, ℱ). Define an equivalence relation on U 퐻 (U, ℱ) as follows: Let ℎ1 ∈ 퐻 (U, ℱ) 푛 ′ and ℎ2 ∈ 퐻 (U , ℱ), we have that ℎ1 ∼ ℎ2 if and only if there exists an open cover B finer that ′ 푝 푝 U and U such that 휏훽 (ℎ1) = 휏훼(ℎ2) where 훽 : 퐼 → Λ and 훼 : 퐽 → Λ. Now, we can take the inductive limit over all the open coverings U and give the next definition. CHAPTER 2. RIEMANN–ROCH THEOREM 29
Definition 2.1.7. Define the 푝th Čech cohomology group as
퐻푝 푋, ℱ 푙푖푚퐻푝 U, ℱ . ( ) = −→ ( ) 푈
Notice that when 푝 = 0, we have that 퐻0(푋, ℱ) = Γ(푋, ℱ).
The next theorem can be found in [14, p. 296]. Theorem 2.1.8. Let ℱ be a sheaf on 푋. Then, 퐻1(푋, ℱ) = 0 if and only if 퐻1(U, ℱ) = 0 for every open cover U of 푋.
Proposition 2.1.9. Let 푋 be a variety and ℱ = 풦(푃 ) be the skyscraper sheaf for some 푃 ∈ 푋. Then 퐻푖(푋, ℱ) = 0 for every 푖 > 0.
′ Proof. Let U = {푈푖}푖∈퐼 be an open cover of 푋. Consider the refinement U of U defined as 푈0 ∪ {푈푖 ∖ {푃 }}푖∈퐼 , where 푃 ∈ 푈0. Then, 푃 is not contained in any intersection of two or more distinct sets in U′. This means that the skyscraper sheaf has no sections over these. Therefore, 퐻푖(푋, ℱ) = 0 for every 푖 > 0.
2.2 Riemann–Roch Theorem
Let 푋 a smooth projective curve, 푃1, ..., 푃푟 points ∈ 푋 and 푎1, ..., 푎푟 ≥ 0. The Riemann–Roch Theorem computes the dimension of the space of rational functions on 푋 that have poles of order at most 푎푖 at the points 푃푖 and are regular everywhere else. 1 Definition 2.2.1. The genus of a smooth curve 푋 is defined as 푔 := dim 퐻 (푋, 풪푋 ). Before starting with the prove of the Riemann–Roch Theorem we need to introduce some theorems of cohomology. These theorems can be found in [8].
Theorem 2.2.2 (Long Exact Sequence of Cohomology). Let 0 → ℱ1 → ℱ2 → ℱ3 → 0 be a short exact sequence of sheaves. Then, there exists a long exact sequence of cohomology groups
0 0 0 0 → 퐻 (푋, ℱ1) → 퐻 (푋, ℱ2) → 퐻 (푋, ℱ3) 1 1 1 → 퐻 (푋, ℱ1) → 퐻 (푋, ℱ2) → 퐻 (푋, ℱ3) 2 → 퐻 (푋, ℱ1) → · · · .
Theorem 2.2.3 (Serre-Duality Theorem). Let 푋 be a projective space of dimension 푛, 퐾 a canonical divisor, and ℱ a locally free sheave. Then for each 0 ≤ 푖 ≤ 푛, there are isomorphisms
퐻푖(푋, ℱ) ∼= 퐻푛−푖(푋, ℒ(퐾) ⊗ ℱ ∨)∨.
Theorem 2.2.4 (Cartan-Serre). Let 푋 be a smooth projective variety, and ℱ be a locally free sheaf. Then, 퐻푛(푋, ℱ) is finite dimensional. CHAPTER 2. RIEMANN–ROCH THEOREM 30
The dimension of the cohomology groups 퐻푖(푋, ℱ) is denoted by ℎ푖(푋, ℱ). Theorem 2.2.5 (Grothendieck’s Vanishing Theorem). Let 푋 be a variety of dimension 푛. For every sheaf of abelian groups ℱ on 푋 we have 퐻푖(푋, ℱ) = 0 for every 푖 > 푛. Definition 2.2.6. Let 푋 be a projective variety over a field 푘, and let ℱ be a locally free sheaf on 푋. The Euler characteristic of ℱ is defined by ∑︁ 풳 (ℱ) = (−1)푖 dim 퐻푖(푋, ℱ). 푖=0
Lemma 2.2.7. Let ℱ, 풢, and ℋ be locally free sheaves. If there is an exact sequence of the form
푓 푓 푓 0 −→1 퐻0(푋, ℱ) −→2 퐻0(푋, 풢) −→3 퐻0(푋, ℋ) 푓 푓 푓 −→4 퐻1(푋, ℱ) −→5 퐻1(푋, 풢) −→6 퐻1(푋, ℋ) → · · · . then the alternating sum dim 퐻0(푋, ℱ) − dim 퐻0(푋, 풢) + dim 퐻0(푋, ℋ) − dim 퐻1(푋, ℱ) + dim 퐻1(푋, 풢) − dim 퐻1(푋, ℋ) + ... + (−1)푛 dim 퐻푛(푋, ℱ) + (−1)푛 dim 퐻푛(푋, 풢) + (−1)푛 dim 퐻푛(푋, ℋ). is equal to zero.
Proof. The exact sequence ends for some 푛 by Theorem 2.2.5. We also have that im(푓1) = {0} = ker(푓3푛+1), and im(푓푖) = ker(푓푖+1) for 푖 ∈ {1, ..., 3푛} by exactness. Applying the rank-nullity theorem gives 0 dim 퐻 (푋, ℱ) = dim ker(푓2) + dim im(푓2) = dim im(푓1) + dim ker(푓3) = dim ker(푓3) 0 dim 퐻 (푋, 풢) = dim ker(푓3) + dim im(푓3) = dim ker(푓3) + dim ker(푓4) 0 dim 퐻 (푋, ℋ) = dim ker(푓4) + dim im(푓4) = dim ker(푓4) + dim ker(푓5) . . 푛 dim 퐻 (푋, 풢) = dim ker(푓3푛−1) + dim im(푓3푛−1) = dim ker(푓3푛−1) + dim ker(푓3푛) 푛 dim 퐻 (푋, ℋ) = dim ker(푓3푛) + dim im(푓3푛) = dim ker(푓3푛) + dim ker(푓3푛+1) = dim ker(푓3푛) Alternately subtracting and adding rows we obtain dim 퐻0(푋, ℱ) − dim 퐻0(푋, 풢) + dim 퐻0(푋, ℋ) − dim 퐻1(푋, ℱ) + dim 퐻1(푋, 풢) − dim 퐻1(푋, ℋ) + ... + (−1)푛 dim 퐻푛(푋, ℱ) + (−1)푛 dim 퐻푛(푋, 풢) + (−1)푛 dim 퐻푛(푋, ℋ) = 0. which finishes the prove.
Proposition 2.2.8. Let 푋 be a smooth projective variety. If
0 → ℱ → 풢 → ℋ → 0 is a short exact sequence of locally free sheaves. Then, 풳 (풢) = 풳 (ℱ) + 풳 (ℋ). CHAPTER 2. RIEMANN–ROCH THEOREM 31
Proof. The sequence 0 → ℱ → 풢 → ℋ → 0 induces a long exact sequence by Theorem 2.2.2.
푓 푓 푓 0 −→1 퐻0(푋, ℱ) −→2 퐻0(푋, 풢) −→3 퐻0(푋, ℋ) 푓 푓 푓 −→4 퐻1(푋, ℱ) −→5 퐻1(푋, 풢) −→6 퐻1(푋, ℋ) → · · · . By Lemma 2.2.7, we obtain that the alternating sum
dim 퐻0(푋, ℱ) − dim 퐻0(푋, 풢) + dim 퐻0(푋, ℋ) − dim 퐻1(푋, ℱ) + dim 퐻1(푋, 풢) − dim 퐻1(푋, ℋ) + ... + (−1)푛 dim 퐻푛(푋, ℱ) + (−1)푛 dim 퐻푛(푋, 풢) + (−1)푛 dim 퐻푛(푋, ℋ) = 0.
The left hand side of this last equation is equal to 풳 (ℱ) − 풳 (풢) + 풳 (ℋ). Therefore, 풳 (ℱ) = 풳 (풢) + 풳 (ℋ).
Theorem 2.2.9 (Riemann–Roch Theorem). Let 푋 be a smooth projective curve of genus 푔. Then for all 퐷 ∈ 퐷푖푣(푋),
푙(퐷) − 푙(퐾푋 − 퐷) = deg(퐷) − 푔 + 1 where 퐾푋 is a canonical divisor of 푋.
0 0 Proof. Remember that 푙(퐷) = dim 퐻 (푋, ℒ(퐷)), and 푙(퐾푋 − 퐷) = dim 퐻 (푋, ℒ(퐾푋 − 퐷)). Applying Theorem 1.2.16, we obtain that
0 0 ∨ 푙(퐾푋 − 퐷) = dim 퐻 (푋, ℒ(퐾푋 − 퐷)) = dim 퐻 (푋, ℒ(퐾푋 ) ⊗ ℒ(퐷) ).
Since 푋 is a proyective variety, we can use the Serre Duality Theorem (2.2.3) to obtain that 0 ∨ 1 퐻 (푋, ℒ(퐾푋 ) ⊗ ℒ(퐷) ) and 퐻 (푋, ℒ(퐷)) are dual. Then,
0 ∨ 1 dim 퐻 (푋, ℒ(퐾푋 ) ⊗ ℒ(퐷) ) = dim 퐻 (푋, ℒ(퐷)).
Therefore, we only need to prove that
dim 퐻0(푋, ℒ(퐷)) − dim 퐻1(푋, ℒ(퐷)) = 1 − 푔 + deg(퐷).
To prove this, we will use induction on deg(퐷), since any divisor is the sum of finite points. i) Case 퐷 = 0(when deg(퐷) = 0). By Theorem 1.2.17 we have that ℒ(0) = 풪푋 . Then, we need to prove that
0 1 dim 퐻 (푋, 풪푋 ) − dim 퐻 (푋, 풪푋 ) = 1 − 푔 + 0.
0 This is true, beacuse 퐻 (푋, 풪푋 ) = 풪푋 (푋) = 푘 which holds since 푋 is a projective variety. 0 1 Therefore, dim 퐻 (푋, 풪푋 ) = 1. Also, dim 퐻 (푋, 풪푋 ) = 푔 by definition of the genus of 푋. ii) Now, assume that the claim is true for any divisor 퐷. We will show that it remains true for 퐷 + 푃 . By Lemma 1.2.21 there is an exact sequence
0 → ℒ(퐷) → ℒ(퐷 + 푃 ) → 풦(푃 ) → 0 CHAPTER 2. RIEMANN–ROCH THEOREM 32 and by Theorem 2.2.2 we obtain a long exact sequence of the form 푓 푓 푓 0 −→1 퐻0(푋, ℒ(퐷)) −→2 퐻0(푋, ℒ(퐷 + 푃 )) −→3 푘 푓 푓 푓 −→4 퐻1(푋, ℒ(퐷)) −→5 퐻1(푋, ℒ(퐷 + 푃 )) −→6 0. 0 1 since dim 퐻 (푋, 풦(푃 )) = dim 푘 = 1, and 퐻 (푋, 풦(푃 )) = 0 by Proposition 2.1.9. Moreover, by Lemma 2.2.7 we obtain dim 퐻0(푋, ℒ(퐷)) − dim 퐻0(푋, ℒ(퐷 + 푃 )) + 1 − dim 퐻1(푋, ℒ(퐷)) + dim 퐻1(푋, ℒ(퐷 + 푃 )) = 0 Also, deg(퐷 + 푃 ) − deg(퐷) = 1. Replacing and rearranging yields dim 퐻0(푋, ℒ(퐷)) − dim 퐻1(푋, ℒ(퐷)) − deg(퐷) = dim 퐻0(푋, ℒ(퐷 + 푃 )) − dim 퐻1(푋, ℒ(퐷 + 푃 )) − deg(퐷 + 푃 ) 0 1 By the induction hypothesis we have that dim 퐻 (푋, ℒ퐷) − dim 퐻 (푋, ℒ퐷) = 1 − 푔 + deg(퐷), so dim 퐻0(푋, ℒ(퐷 + 푃 )) − dim 퐻1(푋, ℒ(퐷 + 푃 )) = 1 − 푔 + deg(퐷 + 푃 ). This finishes the proof.
Corollary 2.2.10. For any canonical divisor 퐾푋 we have that
deg(퐾푋 ) = 2푔 − 2 and 푙(퐾푋 ) = 푔.
Proof. The Riemann–Roch Theorem for 퐷 = 0 gives
푙(0) = deg(0) + 1 − 푔 + 푙(퐾푋 ) by Lemma 1.2.25 (i) we have that 푙(0) = 1, then 푙(퐾푋 ) = 푔. Finally, applying the Riemann–Roch Theorem for 퐷 = 퐾푋 we obtain
푔 = 푙(퐾푋 ) = deg(퐾푋 ) + 1 − 푔 + 푙(퐾푋 − 퐾푋 )
Therefore, deg(퐾푋 ) = 2푔 − 2.
The proof of the following Corollary can be found in [17, p. 216]. Corollary 2.2.11. Let 푋 be a nonsingular curve, then the genus of 푋 is zero, if and only if, 푋 and P1 are isomorphisms. Theorem 2.2.12. If 퐷 is any divisor with deg(퐷) ≥ 2푔 − 1, then 푙(퐷) = deg(퐷) + 1 − 푔.
Proof. By the Riemann–Roch Theorem we have that
푙(퐷) = deg(퐷) − 푔 + 1 + 푙(퐾푋 − 퐷)
Since deg(퐷) ≥ 2푔 − 1 and deg(퐾푋 ) = 2푔 − 2 by corollary 2.2.10 we have that
deg(퐾푋 − 퐷) = deg(퐾푋 ) − deg(퐷) = 2푔 − 2 − deg(퐷) ≤ 2푔 − 2 + 1 − 2푔 = −1
That is, deg(퐾푋 − 퐷) < 0 and by Lemma 1.2.25 we have that 푙(퐾푋 − 퐷) = 0. Therefore, 푙(퐷) = deg(퐷) + 1 − 푔. CHAPTER 2. RIEMANN–ROCH THEOREM 33
Proposition 2.2.13. Let ℰ be a locally free sheaf of rank 푟 over 푋. There exists a short exact sequence of locally free sheaves over 푋
0 → ℒ → ℰ → ℱ → 0 such that ℒ is an invertible sheaf and ℱ has rank 푟 − 1.
Proof. Take an effective divisor 퐷 on 푋 such that 푟 deg(퐷) > ℎ1(푋, ℰ). Consider the short exact sequence 0 → 풪푋 → 풪푋 (퐷) → 풪퐷 → 0 Tensored by ℰ we get 0 → ℰ → ℰ ⊗ 풪푋 (퐷) → ℰ ⊗ 풪퐷 → 0 Then, there is a long exact sequence of cohomology associated to this exact sequence.
0 0 0 0 → 퐻 (푋, ℰ) → 퐻 (푋, ℰ ⊗ 풪푋 (퐷)) → 퐻 (푋, ℰ ⊗ 풪퐷) → 퐻1(푋, ℰ) → · · · .
1 0 1 Since 푟 deg(퐷) > ℎ (푋, ℰ), we have that the map 퐻 (푋, ℰ ⊗ 풪푋 (퐷)) → 퐻 (푋, ℰ) can not be injective. Therefore, we can find a non-zero section of ℰ(퐷) = ℰ ⊗ 풪푋 (퐷), say 푠. This section 푠 defines a map 풪푋 (−퐷) → ℰ, whose image is an invertible sheaf. The degree of an invertible subsheaf ℒ of ℰ is bounded above. In fact, if deg(ℒ) > 2푔 − 2 then 퐻1(푋, ℒ) = 0 by Theorem 2.2.12. Now, applying the Riemann–Roch Theorem we get deg(ℒ) = ℎ0(푋, ℒ) − 1 + 푔 ≤ ℎ0(푋, ℰ) − 1 + 푔, so the deg(ℒ) is bounded above. Take a maximal degree invertible subsheaf ℒ. This guarantees that the cokernel is also locally-free. So we have an exact sequence
0 → ℒ → ℰ → ℰ/ℒ → 0. where ℰ/ℒ has rank 푟 − 1.
Theorem 2.2.14 (Hirzebruch–Riemann–Roch Theorem). Let 푋 be a nonsingular projective curve of genus 푔, and let 퐸 be a vector bundle of rank 푟 on 푋. Then, we have that
ℎ0(푋, 퐸) − ℎ1(푋, 퐸) = deg(퐸) + 푟(1 − 푔).
Proof. The proof will be by induction on the rank 푟 of 퐸. If 푟 = 1, we have the Riemann–Roch Theorem. So, assume that the theorem is true for a vector bundle of rank 푟 − 1. Consider the exact sequence 0 → ℒ → ℰ → ℰ/ℒ → 0 (2.2.1) where ℒ is an invertible sheaf and ℰ/ℒ has rank 푟 − 1. Applying the Riemann–Roch Theorem to the invertible sheaf ℒ we get
풳 (ℒ) = deg(퐿) + 1 − 푔.
Applying the induction hypothesis to the locally free sheaf ℰ/ℒ of rank 푟 − 1 we obtain that
풳 (ℰ/ℒ) = deg(ℰ/ℒ) + (푟 − 1)(1 − 푔). CHAPTER 2. RIEMANN–ROCH THEOREM 34
By Proposition 2.2.8 and the exact sequence (2.2.1) follows that
풳 (ℰ) =풳 (ℒ) + 풳 (ℰ/ℒ) = deg(퐿) + 1 − 푔 + deg(ℰ/ℒ) + (푟 − 1)(1 − 푔)
Since det 퐸 = (det 퐿) · (det 퐸/퐿), we have deg(ℰ) = deg(ℒ) + deg(ℰ/ℒ). Therefore, rearranging and adding
풳 (퐸) = ℎ0(푋, 퐸) − ℎ1(푋, 퐸) = deg(퐸) + 푟(1 − 푔). and the proof is completed.
Proposition 2.2.15. For locally free sheaves ℰ and ℱ over 푋 we have that
deg(ℰ ⊗ ℱ) = rk(ℰ) deg(ℱ) + rk(ℱ) deg(ℰ)
Proof. We are going to prove this by induction on the rank of ℰ. ∼ If rk(ℰ) = 1, then ℰ = 풪푋 (퐷) for some divisor 퐷. If 퐷 is effective consider the exact sequence
0 → 풪푋 → 풪푋 (퐷) → 풪퐷 → 0
Tensoring it with ℱ we obtain
0 → ℱ → 풪푋 (퐷) ⊗ ℱ → 풪퐷 ⊗ ℱ → 0
Therefore, deg(풪푋 (퐷) ⊗ ℱ) = deg(ℱ) + deg(풪퐷 ⊗ ℱ) By the Hirzebruch–Riemann–Roch Theorem we have that
deg(풪퐷 ⊗ ℱ) = 풳 (풪퐷 ⊗ ℱ) − rk(풪퐷 ⊗ ℱ)(1 − 푔).
Thus, deg(풪푋 (퐷) ⊗ ℱ) = deg(ℱ) + 풳 (풪퐷 ⊗ ℱ) − rk(풪퐷 ⊗ ℱ)(1 − 푔)
Since 풳 (푂퐷 ⊗ ℱ) = rk(ℱ)풳 (풪퐷) and rk(풪퐷 ⊗ ℱ) = rk(ℱ) we get
deg(풪푋 (퐷) ⊗ ℱ) = deg(ℱ) + rk(ℱ)풳 (풪퐷) − rk(ℱ)(1 − 푔)
= deg(ℱ) + rk(ℱ)(풳 (풪퐷) − (1 − 푔)) = deg(ℱ) + rk(ℱ)(deg(퐷) + (1 − 푔) − (1 − 푔)) = deg(ℱ) + rk(ℱ)(deg(퐷)).
That is, deg(풪푋 (퐷) ⊗ ℱ) = deg(ℱ) + rk(ℱ)(deg(풪푋 (퐷)). Now, if 퐷 is not effective, we can write 퐷 = 퐷1 − 퐷2 for effective divisors 퐷1, and 퐷2. Then, we have the exact sequences
0 → 풪푋 → 풪푋 (퐷1) → 풪퐷1 → 0 and
0 → 풪푋 → 풪푋 (−퐷2) → 풪−퐷2 → 0. CHAPTER 2. RIEMANN–ROCH THEOREM 35
Therefore,
0 → 풪푋 → 풪푋 (퐷1 − 퐷2) → 풪퐷1−퐷2 → 0 Tensoring it with ℱ we get
0 → ℱ → 풪푋 (퐷1 − 퐷2) ⊗ ℱ → 풪퐷1−퐷2 ⊗ ℱ → 0
By a similar procedure we obtain that
deg(풪푋 (퐷1 − 퐷2) ⊗ ℱ) = deg(ℱ) + rk(ℱ)(deg(풪푋 (퐷1 − 퐷2)).
Now, assume that the formula is true if rk(ℰ) = 푛 − 1, and consider the exact sequence
0 → 풪푋 (퐷) → ℰ → ℰ/풪푋 (퐷) → 0.
Tensoring it with ℱ we get
0 → 풪푋 (퐷) ⊗ ℱ → ℰ ⊗ ℱ → ℰ/풪푋 (퐷) ⊗ ℱ → 0.
Therefore, deg(ℰ ⊗ ℱ) = deg(풪푋 (퐷) ⊗ ℱ) + deg(ℰ/풪푋 (퐷) ⊗ ℱ). (2.2.2)
Since rk(ℰ/풪푋 (퐷)) = 푛 − 1, by the induction hypothesis we have that
deg(ℰ/풪푋 (퐷) ⊗ ℱ) = 푟푘(ℰ/풪푋 (퐷)) deg(ℱ) + rk(ℱ) deg(ℰ/풪푋 (퐷)). (2.2.3) = (rk(ℰ) − 1) deg(ℱ) + rk(ℱ)(deg(ℰ) − deg(풪푋 (퐷))).
Also, deg(풪푋 (퐷) ⊗ ℱ) = deg(ℱ) + rk(ℱ) deg(풪푋 (퐷)) (2.2.4) Replacing equations (2.2.3) and (2.2.4) in (2.2.2) and operating we get
deg(ℰ ⊗ ℱ) = rk(ℰ) deg(ℱ) + rk(ℱ) deg(ℰ).
Proposition 2.2.16. If there is an exact sequence
0 → 퐹 → 퐸 → 퐺 → 0 of vector bundles. Then deg(퐸) = deg(퐹 ) + deg(퐺).
Proof. Applying Proposition 2.2.8 to this exact sequence we get 풳 (퐸) = 풳 (퐹 ) + 풳 (퐺). By the Hirzebruch–Riemman–Roch Theorem we get
deg(퐸) = deg(퐹 ) + rk(퐹 )(1 − 푔) + deg(퐺) + rk(퐺)(1 − 푔) − rk(퐸)(1 − 푔).
Since rk(퐸) = rk(퐹 ) + rk(퐺) we obtain that
deg(퐸) = deg(퐹 ) + deg(퐺). CHAPTER 2. RIEMANN–ROCH THEOREM 36
Proposition 2.2.17. Let 퐸 be a semistable line bundle on 푋 and 퐿 a line bundle on 푋. Then, (i) 휇(퐸 ⊗ 퐿) = 휇(퐸) + 휇(퐿). (ii) 퐸 ⊗ 퐿 is semistable.
Proof. For part (i) we just need to apply the degree formula.
deg(퐸 ⊗ 퐿) deg(퐸) + deg(퐿) rk(퐸) 휇(퐸 ⊗ 퐿) = = = 휇(퐸) + 휇(퐿) rk(퐸 ⊗ 퐿) rk(퐸)
For part (ii), let 퐹 be a subbundle of 퐸 ⊗ 퐿. Remember that 퐿−1 = 퐿∨, then 퐹 ⊗ 퐿∨ is a subbundle of 퐸. Since 퐸 is semistable we have that 휇(퐹 ⊗ 퐿∨) ≤ 휇(퐸). Also, 휇(퐹 ⊗ 퐿∨) = 휇(퐹 ) − deg(퐿) so 휇(퐹 ) ≤ 휇(퐸) + deg(퐿) = 휇(퐸 ⊗ 퐿) and the result follows.
Proposition 2.2.18. Let 퐸 be a semistable bundle on 푋 of slope 휇(퐸) < 0, then 퐻0(푋, 퐸) = 0
Proof. Suppose that 퐻0(푋, 퐸) ≠ 0, then there is a section 푠 ≠ 0 which defines an injection 푠 풪푋 −→ 퐸.
Proposition 2.2.19. Let 퐸 be a semistable bundle on 푋 of slope 휇(퐸) greater than 2푔 − 2, then 퐻1(푋, 퐸) = 0.
Proof. We have that
∨ 휇(퐾푋 ⊗ 퐸 ) =휇(퐾푋 ) − 휇(퐸) =2푔 − 2 − 휇(퐸)
0 ∨ By hypothesis 2푔 − 2 − 휇(퐸) < 0. Then, by Proposition 2.2.18 we get ℎ (푋, 퐾푋 ⊗ 퐸 ) = 0. 1 0 ∨ Applying the Serre Duality Theorem 2.2.3 we obtain ℎ (푋, 퐸) = ℎ (푋, 퐾푋 ⊗ 퐸 ) = 0.
Theorem 2.2.20. Let 퐸 be a semistable bundle. If 휇(퐸) > 2푔 − 2, then
ℎ0(푋, 퐸) = deg(퐸) + rk(퐸)(1 − 푔).
Proof. By Proposition 2.2.19 we have that 퐻1(푋, 퐸) = 0, and the result follows from the Hirzebruch–Riemann–Roch Theorem. 37
Chapter 3
The Weierstrass Semigroup
In this section we will give many ways to generalize the definition of the Weierstrass Semigroup 퐻푃 of a point P.
3.1 The Weierstrass Semigroup of a point P
In this subsection we will define the Weierstrass Semigroup 퐻푃 of a point 푃 in a nonsingular projective curve 푋, and we will prove the Weierstrass Gap Theorem which first appeared in the dissertation of Weierstrass’ student Schottky, and after published as [15]. This theorem says that the cardinality of the complement of 퐻푃 is the genus 푔 of 푋. After that, we will give an example of a Weierstrass point in the Klein quartic. We will denote Z+ = N ∪ {0}. Definition 3.1.1. A Numerical semigroup is a subset 퐻 of Z+ satisfying 1. 0 is an element of 퐻. 2. If 푥 and 푦 are in 퐻 then 푥 + 푦 is also in 퐻. 3. Z+ ∖ 퐻, the complement of 퐻 in Z+, is finite. Example 3.1.2. The set 퐻 = {0, 4, 6, 8, 9, 10, 12, 13, ...} is a numerical semigroup because 0 ∈ 퐻, if 푥 and 푦 are in 퐻 then 푥 + 푦 is also in 퐻, and Z+ − 퐻 = {1, 2, 3, 5, 7, 11} is finite. Definition 3.1.3. Let 퐻 ⊂ Z+ be a numerical semigroup and 퐸 ⊂ Z+ be any set. If 퐻+퐸 ⊂ 퐸 we say that 퐸 is a relative ideal of 퐻. The next definition is the Weierstrass semigroup of apoint 푃 . In Theorem 3.1.8 we will show that this set is in fact a numerical semigroup. + Definition 3.1.4. Let 푋 be a nonsingular curve. For any point 푃 ∈ 푋 define the set 퐻푃 ⊂ Z by + 퐻푃 = {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 }.
+ Let 퐺푃 := Z − 퐻푃 = {푛 : there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 }. The elements 푛 ∈ 퐺푃 are called gap numbers of 푃 . CHAPTER 3. THE WEIERSTRASS SEMIGROUP 38
Lemma 3.1.5. Let 푃 ∈ 푋, and 푛 a non-negative integer, then there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 if and only if ℒ(푛푃 ) = ℒ((푛 − 1)푃 ) .
Proof. Suppose there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 . Since 푛푃 − 푃 ≤ 푛푃 we have that ℒ((푛 − 1)푃 ) ⊂ ℒ(푛푃 ). Now, we need to show that ℒ(푛푃 ) ⊂ ℒ((푛 − 1)푃 ). Suppose there is 푓 ∈ ℒ(푛푃 ) such that 푓∈ / ℒ((푛 − 1)푃 ), i.e. −푛푃 ≤ 푑푖푣(푓) < −(푛 − 1)푃 . Then 푑푖푣(푓) = −푛푃 , i.e. 푣푃 (푓) = −푛 and 푣푄(푓) = 0 for all 푄 ≠ 푃 . Therefore, 푑푖푣(푓)∞ = 푛푃 which is a contradiction.
Reciprocally, if ℒ(푛푃 ) = ℒ((푛 − 1)푃 ), suppose that there is 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 . This is, 푣푃 (푓) = −푛 and 푣푄(푓) ≥ 0 for every 푄 ≠ 푃 . Therefore, 푓 ∈ ℒ(푛푃 ). However, 푓∈ / ℒ((푛−1)푃 ) because 푣푃 (푓) = −푛 < −푛+1 = −푣푃 ((푛−1)푃 ), i.e. 푣푃 (푓)+푣푃 ((푛−1)푃 ) < 0. Consequently, ℒ(푛푃 ) ≠ ℒ((푛 − 1)푃 ) which is a contradiction.
In the next proposition we show that the largest gap number 푛 ∈ 퐺푃 is less than 2푔. Proposition 3.1.6. Let 푃 ∈ 푋. If 푛 is a gap number of 푃 we have that 푛 < 2푔, where 푔 is the genus of 푋.
Proof. Suppose that 푛 ≥ 2푔. Consider the divisors (푛 − 1)푃 and 푛푃 , then
deg((푛 − 1)푃 ) = 푛 − 1 ≥ 2푔 − 1 and deg(푛푃 ) = 푛 > 2푔 − 1
Now, applying Theorem 2.2.12 to (푛 − 1)푃 and 푛푃 we get
푙((푛 − 1)푃 ) = deg((푛 − 1)푃 ) − 푔 + 1 = 푛 − 푔 푙(푛푃 ) = deg(푛푃 ) − 푔 + 1 = 푛 − 푔 + 1
Therefore, 푙((푛 − 1)푃 ) < 푙(푛푃 ), i.e. ℒ((푛 − 1)푃 ) ( ℒ(푛푃 ). By Lemma 3.1.5 there exist 푓 ∈ 푘(푋) such that 푑푖푣(푓)∞ = 푛푃 , then 푛 is not a gap number of 푃 .
The next theorem is the so-called Lückensatz of Weierstrass. It says that the cardinality of 퐺푃 is exactly 푔. Theorem 3.1.7 (Weierstrass Gap Theorem). Let 푋 be a projective nonsingular curve of genus 푔 > 0 and let 푃 be a point of 푋. Then there are exactly 푔 gap numbers 푖1 < ... < 푖푔 of 푃 . We have
푖1 = 1 and 푖푔 ≤ 2푔 − 1. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 39
Proof. If 푛 is a gap of 푃 then 0 ≤ 푛 ≤ 2푔 − 1 by Proposition 3.1.6. Note that 푛 = 0 is not a gap of 푃 because for every constant function 푓 = 푐 ∈ 푘 ⊂ 푘(푋) we have 푑푖푣(푓)∞ = 0푃 .
We affirm that 푛 = 1 is a gap of 푃 . Suppose that 푛 = 1 is not a gap of 푃 , then there exists 푓 ∈ 푘(푋) such that 푑푖푣(푓)∞ = 1푃 , by Theorem 1.2.24 we have that 1 = deg(푑푖푣(푓)∞) = [푘(푋): 푘(푓)], i.e. 푋 and P1 are birrational equivalent. This is a contradiction because the genus of P1 is zero.
Consider now the sequence of vector spaces
ℒ(0) ⊆ ℒ(1푃 ) ⊆ ℒ(2푃 ) ⊆ ... ⊆ ℒ((2푔 − 1)푃 ) (3.1.1)
We know that 푙(0) = 1 and since deg((2푔 − 1)푃 ) = 2푔 − 1 we can apply Theorem 2.2.12 to obtain that 푙((2푔 − 1)푃 ) = deg((2푔 − 1)푃 ) + 1 − 푔 = 푔. Moreover, by Lemma 1.2.26 푙(푖푃 ) − 푙((푖 − 1)푃 ) ≤ 1, this is,
either ℒ(푖푃 ) = ℒ((푖 − 1)푃 ) or 푙(푖푃 ) = 푙((푖 − 1)푃 ) + 1
We have in (3.1.1) 2푔 − 1 consecutive inclusions, but as the dimension of the spaces varies from 1 to 푔, we have that there are 푔 − 1 proper inclusions. Then we have (2푔 − 1) − (푔 − 1) = 푔 equalities. This is, we have exactly 푔 numbers 푖 ∈ {1, ..., 2푔 −1} such that ℒ(푖푃 ) = ℒ((푖−1)푃 ). By Lemma 3.1.5 we have exactly 푔 gaps 푖1 < ... < 푖푔 of 푃 .
Now, we are ready to prove that 퐻푃 is a numerical semigroup. Theorem 3.1.8. Let 푋 be a nonsingular curve of genus 푔. For any point 푃 ∈ 푋, the set + 퐻푃 = {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 } is a numerical semigroup.
Proof.
(i) Constant functions 푓 = 푐 have no poles and satisfy 푣푃 (푓) = 0 for all 푃 ∈ 푋. Then 0 ∈ 퐻푃 .
(ii) Let 푚, 푛 ∈ 퐻푃 then there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 and 푔 ∈ 푘(푋) with 푑푖푣(푔)∞ = 푚푃 . Note that 푑푖푣(푓 ·푔)∞ = 푑푖푣(푓)∞ +푑푖푣(푔)∞ = (푛+푚)푃 . Therefore, there exists 푓 · 푔 ∈ 푘(푋) with 푑푖푣(푓 · 푔)∞ = (푛 + 푚)푃 . Then 푛 + 푚 ∈ 퐻푃 . (iii) By Theorem 3.1.7 we have that the cardinality of
퐺푃 = {푛 : there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛푃 }
is 푔, where 푔 is the genus of 푋. Then 퐺푃 is finite.
Definition 3.1.9. If the gap numbers at 푃 ∈ 푋 are any others than {1, 2, 3, ..., 푔} where 푔 is the genus of 푋, then 푃 is a Weierstrass Point. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 40
Example 3.1.10. The Klein quartic over C , is defined by 퐾 : 푋3푌 + 푌 3푍 + 푍3푋 = 0
푌 Let 푃 = [0 : 0 : 1] ∈ 퐾, 푡 = 푍 is a local parameter at 푃 since 퐹푋 (푃 )푋+퐹푌 (푃 )푌 +퐹푍 (푃 )푍 = 푋 is not a multiple of 푌 .
3 3 3 푋 3 푍 푍 3 푋 Since 푋 푌 + 푌 푍 + 푍 푋 = 0 is equivalent to ( 푌 ) + ( 푌 ) + ( 푌 ) 푌 = 0 we get 푋 푍 3푣푃 ( ) = 푣푃 ( ) or 푌 푌 푋 푍 푋 3푣푃 ( ) = 3푣푃 ( ) + 푣푃 ( ) or 푌 푌 푌 푍 푍 푋 푣푃 ( ) = 3푣푃 ( ) + 푣푃 ( ). 푌 푌 푌 푍 We have that 푣푃 ( 푌 ) = −1, then 푋 3푣푃 ( ) = −1 or 푌 푋 푋 3푣푃 ( ) = −3 + 푣푃 ( ) or 푌 푌 푋 −1 = −3 + 푣푃 ( ). 푌 That is, 푋 푣푃 ( ) = −1/3 or 푌 푋 푣푃 ( ) = −3/2 or 푌 푋 푣푃 ( ) = 2. 푌 푋 The valuation of a function 푓 in a point 푃 is an integer, so we conclude that 푣푃 ( 푌 ) = 2.
3 3 3 푋 3 푌 푌 3 푋 Since 푋 푌 + 푌 푍 + 푍 푋 = 0 is equivalent to ( 푍 ) 푍 + ( 푍 ) + 푍 = 0 we obtain that 푋 푌 푌 3푣푃 ( ) + 푣푃 ( ) =3푣푃 ( ) or 푍 푍 푍 푋 푌 푣푃 ( ) =3푣푃 ( ) or 푍 푍 푋 푌 푋 3푣푃 ( ) + 푣푃 ( ) =푣푃 ( ) 푍 푍 푍 푌 We have that 푣푃 ( 푍 ) = 1, then 푋 3푣푃 ( ) + 1 =3 or 푍 푋 푣푃 ( ) =3 or 푍 푋 푋 3푣푃 ( ) + 1 =푣푃 ( ) 푍 푍 CHAPTER 3. THE WEIERSTRASS SEMIGROUP 41
Then, 푋 푣푃 ( ) =2/3 or 푍 푋 푣푃 ( ) =3 or 푍 푋 푣푃 ( ) = − 1/2 푍
푋 We conclude 푣푃 ( 푍 ) = 3. Now we want to see under which conditions
푖 푗 푌 푍 ⋃︁ 푓푖푗 = 푖+푗 ∈ ℒ(푚푃 ) 푋 푚≥0
We have 푣푃 (푓푖푗) = −2푖 − 3푗 because
푌 푖 푍푗 푌 푍 푣푃 (푓푖푗) = 푣푃 ( ) = 푖푣푃 ( ) + 푗푣푃 ( ) 푋푖 푋푗 푋 푋 = −2푖 − 3푗.
The poles of 푓푖,푗 have 푋 = 0 and it may be at 푃 = [0 : 0 : 1], 푃1 = [0 : 1 : 0]. By a similar procedure we obtain 푌 푣푃 ( ) = −1 1 푋 푍 푣푃 ( ) = 2 1 푋
Then, 푣푃1 (푓푖푗) = −푖 + 2푗. 푌 푖푍푗 ⋃︀ In order to 푓푖푗 = 푋푖+푗 ∈ 푚≥0 ℒ(푚푃 ) we need that −1 + 2푗 ≥ 0 (so they are zeros). Then,
퐻푃 ={2푖 + 3푗 : 푖, 푗 ≥ 0, 2푗 ≥ 푖} ={0, 3, 5, 6, 7, 8, ...}
+ We see that Z − 퐻푃 = {1, 2, 4}, this is, it has 3 gaps. Note that the genus of 퐾 is also 3. Thus, the Weierstrass Gap Theorem 3.1.7 is satisfied. Also, 푃 is a Weierstrass point since his gap sequence is other than {1, 2, 3}.
3.2 The Weierstrass Semigroup of a divisor 퐷
In this section we will define the Weierstrass Semigroup 퐻(퐷) of a divisor 퐷. After that we will find that the largest gap is less than 2푔 where 푔 is the genus of the non singular projective curve 푋. Moreover, in Theorem 3.2.4 we will prove that 퐻(퐷) is in fact a numerical semigroup. Definition 3.2.1. Let 푋 be a non singular projective curve, and 퐷 a divisor on 푋. Then the Weierstrass set of a divisor 퐷 is defined as
+ 퐻(퐷) := {푛 ∈ Z | there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷}. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 42
+ Let 퐺(퐷) := Z − 퐻(퐷) = {푛 : there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷}. The elements 푛 ∈ 퐺(퐷) are called gap numbers of 퐺(퐷).
Note that if we take 퐷 = 푃 we have that this definition coincides with Definition 3.1.4. ∑︀ Lemma 3.2.2. Let 퐷 = 푖 푡푖푃푖 be a divisor on 푋, and 푛 a non-negative integer, then there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷 if and only if ℒ(푛퐷) = ℒ((푛 − 1)퐷) .
Proof. Suppose there is no 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷. Since 푛퐷 − 퐷 ≤ 푛퐷 we have that ℒ((푛 − 1)퐷) ⊂ ℒ(푛퐷). Now, we need to show that ℒ(푛퐷) ⊂ ℒ((푛 − 1)퐷). Suppose there is 푓 ∈ ℒ(푛퐷) such that
푓∈ / ℒ((푛 − 1)퐷), i.e. −푛퐷 ≤ 푑푖푣(푓) < −(푛 − 1)퐷. Then 푑푖푣(푓) = −푛푃 , i.e. 푣푃푖 (푓) = −푛푡푖 for all 푖, and 푣푄(푓) = 0 for all 푄 ≠ 푃푖. Therefore, 푑푖푣(푓)∞ = 푛퐷 which is a contradiction.
Reciprocally, if ℒ(푛퐷) = ℒ((푛 − 1)퐷), suppose that there is 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷.
This is, 푣푃푖 (푓) = −푛푡푖 for all 푖, and 푣푄(푓) ≥ 0 for every 푄 ≠ 푃푖. Therefore, 푓 ∈ ℒ(푛퐷).
However, 푓∈ / ℒ((푛 − 1)퐷) because 푣푃푖 (푓) = −푛푡푖 < −푛푡푖 + 푡푖 = −푣푃푖 ((푛 − 1)퐷), i.e.
푣푃푖 (푓) + 푣푃푖 ((푛 − 1)퐷) < 0 for all 푖. Consequently, ℒ(푛퐷) ≠ ℒ((푛 − 1)퐷) which is a contradiction.
The next proposition states that the largest gap of 퐺(퐷) is less than 2푔. ∑︀ Proposition 3.2.3. Let 퐷 = 푖 푡푖푃푖 be an effective divisor on 푋. If 푛 is a gap number of 퐺(퐷) we have that 푛 < 2푔, where 푔 is the genus of 푋.
Proof. Suppose that 푛 ≥ 2푔. Consider the divisors (푛 − 1)퐷 and 푛퐷, then
deg((푛 − 1)퐷) = (푛 − 1) deg(퐷) ≥ (2푔 − 1) deg(퐷) ≥ 2푔 − 1 where the last inequality is because 퐷 is an effective divisor. Also, deg(푛퐷) = 푛 deg(퐷) > (2푔 − 1) deg(퐷) > 2푔 − 1 where the last inequality is also since 퐷 is an effective divisor. Now, applying Theorem 2.2.12 to (푛 − 1)퐷 and 푛퐷 we get
푙((푛 − 1)퐷) = deg((푛 − 1)퐷) − 푔 + 1 = 푛 deg(퐷) − deg(퐷) − 푔 + 1 푙(푛퐷) = deg(푛퐷) − 푔 + 1 = 푛 deg(퐷) − 푔 + 1
Therefore, 푙((푛 − 1)퐷) < 푙(푛퐷), i.e. ℒ((푛 − 1)퐷) ( ℒ(푛퐷). By Lemma 3.2.2 there exist 푓 ∈ 푘(푋) such that 푑푖푣(푓)∞ = 푛퐷, then 푛 is not a gap number of 퐺(퐷).
In the case that 퐷 is an effective divisor we can prove that 퐻(퐷) has the structure of a numerical semigroup. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 43
∑︀ Theorem 3.2.4. Let 푋 be a nonsingular curve of genus 푔 , and let 퐷 = 푖 푡푖푃푖 be an effective + divisor on 푋. Then the set 퐻(퐷) = {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷} is a numerical semigroup.
Proof.
(i) Constant functions 푓 = 푐 have no poles and satisfy 푣푃푖 (푓) = 0 for all 푃푖 ∈ 푋. Then 0 ∈ 퐻(퐷).
(ii) Let 푚, 푛 ∈ 퐻(퐷) then there exists 푓 ∈ 푘(푋) with 푑푖푣(푓)∞ = 푛퐷 and 푔 ∈ 푘(푋) with 푑푖푣(푔)∞ = 푚퐷. Note that 푑푖푣(푓 · 푔)∞ = 푑푖푣(푓)∞ + 푑푖푣(푔)∞ = (푛 + 푚)퐷. Therefore, there exists 푓 · 푔 ∈ 푘(푋) with 푑푖푣(푓 · 푔)∞ = (푛 + 푚)퐷. Then 푛 + 푚 ∈ 퐻(퐷). (iii) By Proposition 3.2.3 we have that if 푛 ∈ 퐺(퐷), then 푛 < 2푔. That is, 퐺(퐷) is finite.
3.3 The Weierstrass Semigroup of a divisor E with respect to a point P
In this subsection we will define the Weierstrass semigroup 퐻(퐸, 푃 ) of a divisor E with respect to a point P. In Theorem 3.3.6 we will prove that in fact it is a numerical semigroup. We will also compute the cardinality of the complement of 퐻(퐸, 푃 ) in Theorem 3.3.5 which is a theorem analogous to the Weierstrass Gap Theorem for 퐻푃 . Definition 3.3.1. Let 퐸 be a divisor, 푋 a non singular projective curve, and 푃 a point of 푋. Then, the Weierstrass set of a divisor 퐸 with respect to a point 푃 is defined as
+ 퐻(퐸, 푃 ) := {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛푃 }.
+ + Let 퐺(퐸, 푃 ) := Z − 퐻(퐸, 푃 ) = {푛 ∈ Z : there is no 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛푃 }. The elements 푛 of 퐺(퐸, 푃 ) are called gap numbers of 퐺(퐸, 푃 ).
Note that if we take the divisor 퐸 = 0 then this definition is the same as Definition 3.1.4 of the Weierstrass set in a point 푃 . Lemma 3.3.2. 푛 ∈ Z+ is a gap number of 퐺(퐸, 푃 ) if and only if ℒ(퐸+푛푃 ) = ℒ(퐸+(푛−1)푃 ).
Proof. Suppose there is no 푓 ∈ 푘(푋) with 푑푖푣((푓) + 퐸)∞ = 푛푃 . Since 퐸 + (푛 − 1)푃 ≤ 퐸 + 푛푃 we obtain that ℒ(퐸 + (푛 − 1)푃 ) ⊆ ℒ(퐸 + 푛푃 ). Now, we need to show that ℒ(퐸 + 푛푃 ) ⊆ ℒ(퐸 + (푛 − 1)푃 ). Suppose there is 푓 ∈ ℒ(퐸 + 푛푃 ) such that 푓∈ / ℒ(퐸 + (푛 − 1)푃 ). This is −퐸 − 푛푃 ≤ 푑푖푣(푓) < −퐸 − (푛 − 1)푃. Therefore, 푑푖푣(푓) + 퐸 = −푛푃 . That is,
푣푃 ((푓) + 퐸) = −푛,
푣푄((푓) + 퐸) = 0, ∀푄 ≠ 푃. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 44
Then, ((푓) + 퐸)∞ = 푛푃 which is a contradiction.
Reciprocally, if ℒ(퐸 + 푛푃 ) = ℒ(퐸 + (푛 − 1)푃 ), suppose that there is 푓 ∈ 푘(푋) with 푑푖푣((푓) + 퐸)∞ = 푛푃 . This is, 푣푃 ((푓) + 퐸) = −푛, and 푣푄((푓) + 퐸) ≥ 0, ∀푄 ≠ 푃 .Then, 푓 ∈ ℒ(푛푃 + 퐸). However, 푓∈ / ℒ((푛 − 1)푃 + 퐸) because
푣푃 ((푓) + 퐸) = −푛 < −푛 + 1 = −푣푃 ((푛 − 1)푃 )
Consequently, ℒ(퐸 + 푛푃 ) ≠ ℒ(퐸 + (푛 − 1)푃 ) which is a contradiction.
The next proposition states that the largest gap of 퐺(퐸, 푃 ) is less than 2푔 − deg(퐸). Proposition 3.3.3. Let 퐸 be a divisor, 푋 a nonsingular projective curve, and 푃 a point of 푋. If 푛 is a gap number of 퐺(퐸, 푃 ), we have that 푛 < 2푔 − deg(퐸)
Proof. Suppose that 푛 ≥ 2푔 − deg 퐸. Consider the divisors (푛 − 1)푃 + 퐸 and 푛푃 + 퐸, then
deg((푛 − 1)푃 + 퐸) = (푛 − 1) deg(푃 ) + deg(퐸) ≥ (2푔 − deg(퐸) − 1) + deg(퐸) = 2푔 − 1
Furthermore,
deg(푛푃 + 퐸) = 푛 + deg(퐸) ≥ 2푔 − deg(퐸) + deg(퐸) > 2푔 − 1
Applying Theorem 2.2.12 to (푛 − 1)푃 + 퐸 and 푛푃 + 퐸 we get
푙((푛 − 1)푃 + 퐸) = deg((푛 − 1)푃 + 퐸) + 1 − 푔 = 푛 − 1 + deg(퐸) + 1 − 푔. 푙(푛푃 + 퐸) = deg(푛푃 + 퐸) + 1 − 푔 = 푛 + deg(퐸) + 1 − 푔. Therefore, 푙((푛 − 1)푃 + 퐸) < 푙(푛푃 + 퐸), i.e. ℒ((푛 − 1)푃 + 퐸) ( ℒ(푛푃 + 퐸). By Lemma 3.3.2 there exist 푓 ∈ 푘(푋) such that 푑푖푣(푓 + 퐸)∞ = 푛푃 , then 푛 is not a gap number of 퐺(퐸, 푃 ).
Note that if in Proposition 3.3.3 we have 퐸 = 0 , then 푛 < 2푔 as expected. Corollary 3.3.4. If deg(퐸) ≥ 2푔 − 1, then 퐺(퐸, 푃 ) = ∅. The next theorem is an analogous to the Weiertrass Gap Theorem, it tell us that the cardinality of the set 퐺(퐸, 푃 ) is exactly ℓ(퐾푋 − 퐸). Theorem 3.3.5. Let 푋 be a projective non singular curve of genus 푔 > 0, 퐸 a principal divisor such that deg(퐸) < 2푔 − 1, and 푃 a point of 푋. Then there are exactly
ℎ := 푔 + ℓ(퐸) − deg(퐸) − 1 = ℓ(퐾푋 − 퐸) gap numbers 푖1 < ... < 푖ℎ of 퐺(퐸, 푃 ). CHAPTER 3. THE WEIERSTRASS SEMIGROUP 45
Proof. Since 퐸 is a principal divisor, suppose that 퐸 = (휓) for some 휓 ∈ 푘(푋).
Note that 푛 = 0 is not a gap number of 퐺(퐸, 푃 ). Take (푓) = −퐸 = −(휓), then ((푓) + 퐸)∞ = (−(휓) + (휓))∞ = 0푃.
If 푛 is a gap of 퐺(퐸, 푃 ) then 0 < 푛 ≤ 2푔 − deg(퐸) − 1 by Proposition 3.3.3. Consider now the sequence of vector spaces ℒ(퐸 + 0푃 ) ⊆ ℒ(퐸 + 1푃 ) ⊆ ... ⊆ ℒ(퐸 + (2푔 − deg(퐸) − 1)푃 ) (3.3.1) By the Riemann–Roch Theorem
푙(퐸 + 0푃 ) = deg(퐸 + 0푃 ) − 푔 + 1 + 푙(퐾푋 − 퐸 − 0푃 ) (3.3.2) = deg(퐸) − 푔 + 1 + 푙(퐾푋 − 퐸). Also, 푙(퐸 + (2푔 − deg(퐸) − 1)푃 ) = deg(퐸 + (2푔 − deg(퐸) − 1)푃 ) − 푔 + 1
+ 푙(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) (3.3.3)
= 푔 + 푙(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ). Moreover, by Lemma 1.2.26 푙(퐸 + 푖푃 ) − 푙(퐸 + 푖푃 − 푃 ) ≤ 1, that is either ℒ(퐸 + 푖푃 ) = ℒ(퐸 + (푖 − 1)푃 ) or 푙(퐸 + 푖푃 ) = 푙(퐸 + (푖 − 1)푃 ) + 1 As the dimension of the spaces in (3.3.1) varies from 푙(퐸 + 0) to 푙(퐸 + (2푔 − deg(퐸) − 1)푃 ), we have that there are 푙(퐸 + (2푔 − deg(퐸) − 1)푃 ) − 푙(퐸 + 0) proper inclusions. Then, we obtain ℎ = (2푔 − deg(퐸) − 1) − [푙(퐸 + (2푔 − deg(퐸) − 1)푃 ) − 푙(퐸 + 0)] equalities. Replacing equations (3.3.2) and (3.3.3) in the formula for ℎ and operating yields
ℎ = 푙(퐾푋 − 퐸) − 푙(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) (3.3.4)
Note that if in the Riemann-Roch Theorem we replace 퐷 by 퐾푋 − 퐷 we obtain that 푙(퐾푋 − 퐷) = deg(퐾푋 − 퐷) − 푔 + 1 + 푙(퐷). Using this, we have that
푙(퐾푋 − 퐸) = deg(퐾푋 − 퐸) − 푔 + 1 + 푙(퐸), and
푙(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) = deg(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) − 푔 + 1 + 푙(퐸 + (2푔 − deg(퐸) − 1)푃 ). By Theorem 2.2.12 we have that 푙(퐸+(2푔−deg(퐸)−1)푃 ) = deg(퐸)+2푔−deg(퐸)−1−푔+1 = 푔, because deg(퐸 + (2푔 − deg(퐸) − 1)푃 ) = deg(퐸) + 2푔 − deg(퐸) − 1 = 2푔 − 1. Then,
푙(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) = deg(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) + 1. Therefore,
ℎ = deg(퐾푋 − 퐸) − 푔 + 1 + 푙(퐸) − deg(퐾푋 − 퐸 − (2푔 − deg(퐸) − 1)푃 ) − 1 = 푔 + 푙(퐸) − deg(퐸) − 1
= 푙(퐾푋 − 퐸). CHAPTER 3. THE WEIERSTRASS SEMIGROUP 46
So, by Lemma 3.3.2 we have exactly ℎ = 푔 + 푙(퐸) − deg(퐸) − 1 = 푙(퐾푋 − 퐸) gap numbers of 퐺(퐸, 푃 ).
Note that if 퐸 = 0 in Theorem 3.3.5, then ℎ = 푔 + 푙(0) − deg(0) − 1 = 푔 + 1 − 0 − 1 = 푔 as expected from Theorem 3.1.7. In the case that 퐸 is a principal, effective divisor we can prove that 퐻(퐸, 푃 ) has the structure of a numerical semigroup. Theorem 3.3.6. Let 퐸 be a principal, effective divisor, 푋 a nonsingular projective curve of genus 푔, and 푃 a point of 푋. Then the set
+ 퐻(퐸, 푃 ) = {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛푃 } is a numerical semigroup.
Proof.
(i) We prove in Theorem 3.3.5 that 0 is not a gap number of 퐺(퐸, 푃 ). Therefore, 0 ∈ 퐻(퐸, 푃 ).
(ii) Since 퐸 is a principal divisor, let 퐸 = (휓) for some 휓 ∈ 푘(푋). If 푛1 ∈ 퐻(퐸, 푃 ), then there is 푓 ∈ 푘(푋) with ((푓) + (휓))∞ = 푛1푃 , i.e. 푣푃 ((푓) + (휓)) = −푛1. Also, if 푛2 ∈ 퐻(퐸, 푃 ), then there is 푔 ∈ 푘(푋) with ((푔) + (휓))∞ = 푛2푃 , i.e. 푣푃 ((푔) + (휓)) = −푛2. Note that 푣푃 (((푓 + 휓)(푔 + 휓))) ≠ 푣푃 (퐸) since the first one is negative and the other positive. In fact,
푣푃 (((푓 + 휓)(푔 + 휓)) + 퐸) = 푚푖푛{푣푃 (((푓 + 휓)(푔 + 휓))), 푣푃 (퐸)}
= 푚푖푛{푣푃 ((푓 + 휓)) + 푣푃 ((푔 + 휓)), 푣푃 (퐸)}
Since 푣푃 ((푓) + (휓)) + 푣푃 ((푔) + (휓)) = −(푛1 + 푛2) which is a negative number, and 푣푃 (퐸) > 0 for every 푃 ∈ 푋 we obtain that 푣푃 (((푓 + 휓)(푔 + 휓)) + 퐸) = −(푛1 + 푛2). That is, there exist (푓 + 휓)(푔 + 휓) ∈ 푘(푋) such that (((푓 + 휓)(푔 + 휓)) + 퐸)∞ = (푛1 + 푛2)푃 . + (iii) By Theorem 3.3.5 we have that 퐺(퐸, 푃 ) = Z ∖퐻(퐸, 푃 ) has exactly 푙(퐾푋 −퐸) elements. Then, 퐺(퐸, 푃 ) is finite.
3.4 The Weierstrass Semigroup of a divisor 퐸 with respect to a divisor 퐷
In this subsection we will define the Weierstrass Semigroup 퐻(퐸, 퐷) of a divisor 퐸 with respect to a divisor 퐷. Also, we will show in Proposition 3.4.3 that if 푛 is an element of the complement of 퐻(퐸, 퐷), then 푛 < 2푔 − deg(퐸)/deg(퐷). After that we will prove in Theorem 3.4.4 that 퐻(퐸, 퐷) is a numerical semigroup if 퐸 is an effective, principal divisor and 퐷 is an effective divisor. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 47
∑︀ Definition 3.4.1. Let 퐸 be a divisor, 푋 a non singular projective curve, and let 퐷 = 푖 푡푖푃푖 be a divisor on 푋. Then, the Weierstrass set of a divisor 퐸 with respect to 퐷 is defined as
+ 퐻(퐸, 퐷) := {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛퐷}.
+ + Let 퐺(퐸, 퐷) := Z − 퐻(퐸, 퐷) = {푛 ∈ Z : there is no 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛퐷}. The elements 푛 of 퐺(퐸, 퐷) are called gap numbers of 퐺(퐸, 퐷).
Note that if we take the divisor 퐷 = 푃 then this definition is the same as Definition 3.3.1 of the Weierstrass set of a divisor 퐸 with respect to 푃 . ∑︀푟 + Lemma 3.4.2. Let 퐸 be a divisor, and let 퐷 = 푖 푡푖푃푖 be a divisor on 푋. Then, 푛 ∈ Z is a gap number of 퐺(퐸, 퐷) if and only if ℒ(퐸 + 푛퐷) = ℒ(퐸 + (푛 − 1)퐷).
Proof. Suppose there is no 푓 ∈ 푘(푋) with 푑푖푣((푓) + 퐸)∞ = 푛퐷. Since 퐸 + (푛 − 1)퐷 ≤ 퐸 + 푛퐷 we obtain that ℒ(퐸 + (푛 − 1)퐷) ⊆ ℒ(퐸 + 푛퐷). Now, we need to show that ℒ(퐸 + 푛퐷) ⊆ ℒ(퐸 + (푛 − 1)퐷). Suppose there is 푓 ∈ ℒ(퐸 + 푛퐷) such that 푓∈ / ℒ(퐸 + (푛 − 1)퐷). This is
−퐸 − 푛퐷 ≤ 푑푖푣(푓) < −퐸 − (푛 − 1)퐷.
Therefore, 푑푖푣(푓) + 퐸 = −푛퐷. That is,
푣푃푖 ((푓) + 퐸) = −푛푡푖, 푖 = 1, ..., 푟.
푣푄((푓) + 퐸) = 0, ∀푄 ≠ 푃푖.
Then, ((푓) + 퐸)∞ = 푛퐷 which is a contradiction.
Reciprocally, if ℒ(퐸 + 푛퐷) = ℒ(퐸 + (푛 − 1)퐷), suppose that there is 푓 ∈ 푘(푋) with 푑푖푣((푓) + 퐸)∞ = 푛퐷.
This is, 푣푃푖 ((푓) + 퐸) = −푛푡푖, 푖 = 1, ..., 푟, and 푣푄((푓) + 퐸) ≥ 0, ∀푄 ≠ 푃푖.Then, 푓 ∈ ℒ(푛퐷 + 퐸). However, 푓∈ / ℒ((푛 − 1)퐷 + 퐸) because
푣푃푖 ((푓) + 퐸) = −푛푡푖 < −푛푡푖 + 푡푖 = −푣푃푖 ((푛 − 1)퐷).
Consequently, ℒ(퐸 + 푛퐷) ≠ ℒ(퐸 + (푛 − 1)퐷) which is a contradiction.
The next proposition states that the largest gap of 퐺(퐸, 퐷) is less than 2푔 − deg(퐸)/deg(퐷). Proposition 3.4.3. Let 퐸 be a divisor, and 퐷 be an effective divisor on 푋. If 푛 is a gap number of 퐺(퐸, 퐷) we have that 푛 < 2푔 − deg(퐸)/deg(퐷).
Proof. Suppose that 푛 ≥ 2푔 −deg(퐸)/deg(퐷). Consider the divisors (푛−1)퐷 +퐸 and 푛퐷 +퐸, then deg((푛 − 1)퐷 + 퐸) = (푛 − 1) deg(퐷) + deg(퐸) ≥ (2푔 − deg(퐸)/deg(퐷) − 1) deg(퐷) + deg(퐸) = (2푔 − 1) deg(퐷) CHAPTER 3. THE WEIERSTRASS SEMIGROUP 48
Since 퐷 is an effective divisor, we have that deg((푛 − 1)퐷 + 퐸) ≥ 2푔 − 1. Furthermore,
deg(푛퐷 + 퐸) = 푛 deg(퐷) + deg(퐸) > (2푔 − deg(퐸)/deg(퐷)) deg(퐷) + deg(퐸) = 2푔 deg(퐷) > 2푔 deg(퐷) − deg(퐷) = (2푔 − 1) deg(퐷).
Since 퐷 is an effective divisor, we have that deg(푛퐷 + 퐸) ≥ 2푔 − 1. Applying Theorem 2.2.12 to (푛 − 1)퐷 + 퐸 and 푛퐷 + 퐸 we get
푙((푛 − 1)퐷 + 퐸) = deg((푛 − 1)퐷 + 퐸) + 1 − 푔 = (푛 − 1) deg(퐷) + deg(퐸) + 1 − 푔. 푙(푛퐷 + 퐸) = deg(푛퐷 + 퐸) + 1 − 푔 = 푛 deg(퐷) + deg(퐸) + 1 − 푔. Therefore, 푙((푛 − 1)퐷 + 퐸) < 푙(푛퐷 + 퐸), i.e. ℒ((푛 − 1)퐷 + 퐸) ( ℒ(푛퐷 + 퐸). By Lemma 3.4.2 there exist 푓 ∈ 푘(푋) such that 푑푖푣(푓 + 퐸)∞ = 푛퐷, then 푛 is not a gap number of 퐺(퐸, 퐷).
In the case 퐸 is a principal, effective divisor, and 퐷 an effective divisor we can prove that 퐻(퐸, 퐷) has the structure of a numerical semigroup. Theorem 3.4.4. Let 퐸 be a principal, effective divisor, 푋 a nonsingular projective curve of ∑︀푟 genus 푔, and 퐷 = 푖=1 푡푖푃푖 an effective divisor of 푋. Then the set
+ 퐻(퐸, 퐷) = {푛 ∈ Z : there exists 푓 ∈ 푘(푋) with ((푓) + 퐸)∞ = 푛퐷} is a numerical semigroup.
Proof.
(i) Since 퐸 is a principal divisor, suppose that 퐸 = (휓) for some 휓 ∈ 푘(푋). Take (푓) = −퐸 = −(휓), then ((푓) + 퐸)∞ = (−(휓) + (휓))∞ = 0퐷. That is, 0 ∈ 퐻(퐸, 퐷).
(ii) Let 퐸 = (휓) for some 휓 ∈ 푘(푋). If 푛1 ∈ 퐻(퐸, 퐷), then there is 푓 ∈ 푘(푋) with
((푓) + (휓))∞ = 푛1퐷, i.e. 푣푃푖 ((푓) + (휓)) = −푛1푡푖, ∀푖 = 1, ..., 푟. Also, if 푛2 ∈ 퐻(퐸, 퐷),
then there is 푔 ∈ 푘(푋) with ((푔) + (휓))∞ = 푛2퐷, i.e. 푣푃푖 ((푔) + (휓)) = −푛2푡푖, ∀푖 = 1, ..., 푟.
Note that 푣푃푖 (((푓 + 휓)(푔 + 휓))) ≠ 푣푃푖 (퐸) since the first one is negative an the other positive. In fact,
푣푃푖 (((푓 + 휓)(푔 + 휓)) + 퐸) = 푚푖푛{푣푃푖 (((푓 + 휓)(푔 + 휓))), 푣푃푖 (퐸)}
= 푚푖푛{푣푃푖 ((푓 + 휓)) + 푣푃푖 ((푔 + 휓)), 푣푃푖 (퐸)}, ∀푖 = 1, ..., 푟.
Since 푣푃푖 ((푓) + (휓)) + 푣푃푖 ((푔) + (휓)) = −(푛1 + 푛2)푡푖 which is a negative number, and
푣푃푖 (퐸) > 0 for every 푃푖 ∈ 푋 we obtain that 푣푃푖 (((푓 + 휓)(푔 + 휓)) + 퐸) = −(푛1 + 푛2)푡푖. That is, there exist (푓 +휓)(푔+휓) ∈ 푘(푋) such that (((푓 +휓)(푔+휓))+퐸)∞ = (푛1 +푛2)퐷. (iii) By Proposition 3.4.3 we have that if 푛 ∈ 퐺(퐸, 퐷) then 푛 < 2푔 − deg(퐸)/deg(퐷). Thus, 퐺(퐸, 퐷) is finite. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 49
3.5 The Weierstrass set of ℱ with respect to a point 푃
In this subsection we will define the Weierstrass set 푆(ℱ, 푃 ) of a locally free sheaf ℱ with respect to a point P. In Theorem 3.5.3 we will prove that 푆(ℱ, 푃 ) is an 퐻푃 -relative ideal. We will also prove in Theorem 3.5.4 that if 퐹 is semistable then the largest gap of the complement of 푆(ℱ, 푃 ) is less than 2푔 − deg(퐹 )/rk(퐹 ).
Definition 3.5.1. Let ℱ be a sheaf of 풪푋 - modules, and 퐷 a divisor on 푋. The Weierstrass set of ℱ with respect to 퐷 is defined by
+ 0 0 푆(ℱ, 퐷) := {푛 ∈ Z |ℎ (ℱ(푛 − 1)퐷) < ℎ (ℱ(푛퐷))} where ℱ(푛퐷) := ℱ ⊗ 풪푋 (푛퐷). Note that + 퐺(ℱ, 퐷) :=Z ∖ 푆(ℱ, 퐷) + 0 0 ={푛 ∈ Z |ℎ (ℱ(푛퐷)) − ℎ (ℱ(푛 − 1)퐷) ≤ 0} + 0 0 ={푛 ∈ Z |ℎ (ℱ(푛퐷)) = ℎ (ℱ(푛 − 1)퐷)} The elements 푛 of 퐺(ℱ, 퐷) are called the gap numbers of 퐺(ℱ, 퐷).
In the case when 퐷 = 푃 we have the following definition.
Definition 3.5.2. Let ℱ be a sheaf of 풪푋 - modules, and 푃 a prime divisor on 푋. The Weierstrass set of ℱ with respect to 푃 is defined by
+ 0 0 푆(ℱ, 푃 ) := {푛 ∈ Z |ℎ (ℱ(푛 − 1)푃 ) < ℎ (ℱ(푛푃 ))} where ℱ(푛푃 ) := ℱ ⊗ 풪푋 (푛푃 ). ∼ If ℱ is an invertible sheaf, then ℱ = 풪푋 (퐸) for some divisor 퐸. Therefore, ∼ ∼ ℱ(푛퐷) := ℱ ⊗ 풪푋 (푛퐷) = 풪푋 (퐸) ⊗ 풪푋 (푛퐷) = 풪푋 (퐸 + 푛퐷). (3.5.1) In the case when ℱ is an invertible sheaf(line bundle) by Lemma 3.4.2 and equation (3.5.1) we obtain that Definition 3.5.1 is the same as Definition 3.4.1 of the Weierstrass semigroup 퐻(퐸, 퐷) of a divisor E with respect to D. Similarly, if ℱ is a invertible sheaf(line bundle) and 퐷 = 푃 , then by Lemma 3.3.2 and equation (3.5.1) we obtain that Definition 3.5.2 is the same as Definition 3.3.1 of the Weierstrass semigroup 퐻(퐸, 푃 ) of a divisor E with respect to P.
Theorem 3.5.3. Let ℱ be a sheaf of 풪푋 - modules, and 푃 a prime divisor on 푋. Then 푆(ℱ, 푃 ) is an 퐻푃 -relative ideal.
Proof. We need to check that the action
퐻푃 × 푆(ℱ, 푃 ) → 푆(ℱ, 푃 ) (푠, 푒) → 푠 + 푒 is well defined. Take 푓 ∈ 퐻0(푋, ℒ(푠푃 )) ∖ 퐻0(푋, ℒ((푠 − 1)푃 )) and also 푔 ∈ 퐻0(푋, ℱ(푒푃 )) ∖ 퐻0(푋, ℱ((푒 − 1)푃 )). We obtain that 푓푔 ∈ 퐻0(푋, ℱ((푠 + 푒)푃 )), because if there is ℎ ∈ 퐻0(푋, ℒ(푃 )) such that either 푓푔 = ℎ푓 ′푔 for some 푓 ′ ∈ 퐻0(푋, ℒ((푠 − 1)푃 )), or 푓푔 = ℎ푓푔′ for some 푔′ ∈ 퐻0(푋, ℱ((푒 − 1)푃 )), we would find a contradiction with our hypothesis. CHAPTER 3. THE WEIERSTRASS SEMIGROUP 50
deg(퐹 ) The next theorem states that the largest gap of 퐺(퐹, 푃 ) is less than 2푔 − rk(퐹 ) . Theorem 3.5.4. Let 퐹 be a semistable bundle and 푃 a prime divisor on 푋. If 푛 is a gap deg(퐹 ) number of 퐺(퐹, 푃 ), then 푛 < 2푔 − 휇(퐹 ) = 2푔 − rk(퐹 ) .
deg(퐹 ) Proof. Suppose that 푛 ≥ 2푔 − rk(퐹 ) . We have that
⊗푛 deg(퐹 ⊗ 풪푋 (푃 )) rk(퐹 ) · 푛 + deg(퐹 ) rk(퐹 ) · 푛 + deg(퐹 ) ⊗푛 = ⊗푛 = rk(퐹 ⊗ 풪푋 (푃 )) rk(퐹 ) · rk(풪푋 (푃 ))) rk(퐹 ) deg(퐹 ) =푛 + rk(퐹 ) deg(퐹 ) deg(퐹 ) ≥2푔 − + > 2푔 − 2. rk(퐹 ) rk(퐹 )
Also,
deg(퐹 ⊗ 풪⊗(푛−1)(푃 )) rk(퐹 ) · (푛 − 1) + deg(퐹 ) rk(퐹 ) · (푛 − 1) + deg(퐹 ) 푋 = = ⊗(푛−1) ⊗(푛−1) 퐹 rk(퐹 ⊗ 풪푋 (푃 )) rk(퐹 ) · rk(풪푋 (푃 ))) rk( ) deg(퐹 ) =(푛 − 1) + rk(퐹 ) deg(퐹 ) deg(퐹 ) ≥2푔 − − 1 + > 2푔 − 2. rk(퐹 ) rk(퐹 )
⊗(푛−1) ⊗푛 Applying Theorem 2.2.20 to 퐹 ⊗ 풪푋 (푃 ) and 퐹 ⊗ 풪푋 (푃 ) we get
0 ⊗푛 ⊗푛 ⊗푛 ℎ (푋, 퐹 ⊗ 풪푋 (푃 )) = deg(퐹 ⊗ 풪푋 (푃 )) + rk(퐹 ⊗ 풪푋 (푃 ))(1 − 푔) =푛 rk(퐹 ) + deg(퐹 ) + rk(퐹 )(1 − 푔). 0 ⊗(푛−1) ⊗(푛−1) ⊗(푛−1) ℎ (푋, 퐹 ⊗ 풪푋 (푃 )) = deg(퐹 ⊗ 풪푋 (푃 )) + rk(퐹 ⊗ 풪푋 (푃 ))(1 − 푔) = (푛 − 1) rk(퐹 ) + deg(퐹 ) + rk(퐹 )(1 − 푔).
Then,
0 ⊗푛 0 ⊗(푛−1) ℎ (푋, 퐹 ⊗ 풪푋 (푃 )) − ℎ (푋, 퐹 ⊗ 풪푋 (푃 )) =푛 rk(퐹 ) + deg(퐹 ) + rk(퐹 )(1 − 푔) − [(푛 − 1) rk(퐹 ) + deg(퐹 ) + rk(퐹 )(1 − 푔)] = rk(퐹 ) > 0
Therefore, 푛 is not a gap number of 퐺(퐹, 푃 ), i.e. 푛 is a gap number of 퐺(퐹, 푃 ) if 푛 < 2푔 − 휇(퐹 ). ∼ Note that if 퐹 is a line bundle then 퐹 = 풪푋 (퐸) for some divisor 퐸, and the largest gap is less than 2푔 − deg(퐸) as expected from Proposition 3.3.3. 51
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