• algebraic stacks
• D-M stacks
• elliptic curves
• hyperelliptic curves
• Picard groups

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.