Topics in Algebraic Curves: Zeta Function and Frobenius nonclassical curves
Weierstrass points on curves over finite fields and applications
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Author(s): |
Brady Miliwska Ali Medina
Total Authors: 1
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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
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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 |