O teorema de Riemann-Roch e diferentes formas de generalizar o semigrupo de Weierstrass

Author(s)

Brady Miliwska Ali Medina

• Document type: Master's Dissertation
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2020-03-06

Examining board members

Marcos Benevenuto Jardim; Fernando Eduardo Torres Orihuela; Ethan Cotterill

• Advisor: Marcos Benevenuto Jardim

Abstract

The objective of this thesis is to prove the Riemann--Roch Theorem for a smooth projective curve $X$, and to give different ways to generalize the concept of a Weierstrass semigroup $H_P$ of a point $P$ in $X$. We begin by defining the Weierstrass semigroup $H(D)$ of a divisor $D$ and we get that the largest gap is less than $2g$. Then, we define the Weierstrass semigroup $H(E,P)$ of a divisor $E$ with respect to a point $P$ and we obtain that the cardinality of the set of gaps is $l(K_X-E)$. Afterwards, we define the Weierstrass semigroup $H(E,D)$ of a divisor $E$ with respect to $D$ and we have that the largest gap is less than $2g-\deg(E)/\deg(D)$. Finally, we define the Weierstrass set $S(\mathcal{F},P)$ of a vector bundle $\mathcal{F}$ with respect to $P$ and we prove that $S(\mathcal{F},P)$ is an $H_P$-ideal. Furthermore, if $\mathcal{F}$ is semistable then we prove that the largest gap is less than $2g-\deg(\mathcal{F})/\rk(\mathcal{F})$ (AU)

• FAPESP's process: 18/12888-0 - Algebraic curves and the Riemann-Roch theorem
• Grantee: Brady Miliwska Ali Medina
• Support Opportunities: Scholarships in Brazil - Master