UC Berkeley

In this thesis, I study the geometry of algebraic stacks and some important cohomological invariants on them, motivated by moduli-theoretic problems. The language of stacks provides a natural framework for understanding moduli problems that cannot be captured by schemes alone. The first main chapter is based on "Picard Groups of Stacky Curves," and investigates the geometry and Picard groups of stacky curves and gerbes. I further develop the theory of rigidification and show that every stacky curve can be written as a gerbe over a stacky curve with trivial generic stabilizer. I calculate the Picard groups of tame stacky curves with trivial generic stabilizers and express the Picard groups of gerbes as an extension of two groups, which depend on the Brauer classes of the gerbes and the Picard groups of the base. These results together give a description of the Picard group of any tame stacky curve as an extension of two groups. I apply the theory to many examples, including some moduli stacks of interest. The second main chapter is based on "The Brauer Group of BG and Gerbe Structures of Moduli Spaces." I calculate the Brauer group of the classifying stack BG for G a smooth geometrically connected linear algebraic group, and study Brauer-Severi stacks on BG. I calculate the Brauer classes of Brauer-Severi stacks coming from certain representations of the universal cover of G. These results are then applied to study the moduli stack of curves of genus g together with an order N automorphism. In particular, I look at curves whose quotients by the order N automorphism are genus 0, and completely determine the Brauer classes of these gerbes.