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].