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Tesi etd-09102020-102635


Thesis type
Tesi di laurea magistrale
Author
PORZIO, MORENA
URN
etd-09102020-102635
Title
Picard groups of stacks of elliptic and hyperelliptic curves
Struttura
MATEMATICA
Corso di studi
MATEMATICA
Supervisors
relatore Prof. Vistoli, Angelo
Parole chiave
  • hyperelliptic curves
  • elliptic curves
  • D-M stacks
  • algebraic stacks
  • Picard groups
Data inizio appello
25/09/2020;
Consultabilità
Secretata d'ufficio
Riassunto analitico
The purpose of this thesis is to study the geometry of some famous stacks of curves by presenting them as quotient stacks and computing their Picard groups. We start by proving a criterion for an algebraic stack to be a quotient stack. As applications, we see the description as quotient stacks of the stack Mg of smooth curves of genus g, of the stack M1,1 of elliptic curves, and finally of Hg the closed substack of Mg classifying hyperelliptic curves, the last one in characteristic other than 2. First computed by Mumford over a field k of char(k) other than 2,3 in the enlightening paper "Picard Groups of Moduli Problems", we reprise the computation of Picard group of M1,1 over a normal ring, as done in "The Picard Group of M1,1" by Fulton and Olsson. Inspired by the techniques of this paper, we extend the result, contained in the paper "Stacks of Cyclic Covers of Projective Spaces" of Arsie and Vistoli, about Picard groups of Hg, computing them over a normal domain, away from bad characteristics.
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