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The Brauer group of $BG$ and gerbe structures of moduli spaces

Rose Lopez

TL;DR

This work determines the cohomological Brauer group of the classifying stack $BG$ for a smooth connected semisimple group $G$ by identifying $\mathrm{Br}(BG) \cong \widehat{B}$, where $B$ is the fundamental group of $G$ and $\widehat{B}=\mathrm{Hom}(B,\mathbb{G}_m)$. It then analyzes Brauer-Severi stacks over $BG$ and shows that the Brauer class of a stack of the form $[\mathbb{P}V/G] \to BG$ is given by $\chi^{-1}_*(B\widetilde{G})$, where $B$ acts on $V$ via a character $\chi$. Building on this, the paper studies μ_N-gerbes arising from the inertia of the moduli stack of genus $g$ curves, focusing on components with genus-0 quotients; for these, it relates the Brauer class to a parity condition on the data $A=(0,N,d_1,\dots,d_{N-1})$ describing ramification. The main results show that the Brauer group $H^2(\overline{\mathscr N_A}, \mathbb{G}_m)$ is either $\mathbb{Z}/2\mathbb{Z}$ or $0$ depending on parity, and that the Brauer class of $\overline{\mathscr M_A}$ is the pullback $\gamma^*(\mathscr G_{d/N})$, with $d/N$ determining triviality; when $d$ is even the class can be described via a product of simpler gerbes. The approach combines a detailed computation of Br$(BG)$, a stack-theoretic interpretation of Brauer-Severi varieties, rigidification techniques, and Gabber-type arguments to transfer local to global Brauer data on the moduli stacks.

Abstract

We study the $μ_N$-gerbe of curves of genus $g$ with an order $N$ automorphism, and explore what corresponding $H^2$-cohomology classes the components of this stack can have. In particular, we look at curves whose quotients by the order $N$ automorphism are genus 0, and completely determine the Brauer classes of these gerbes. The key technical input is the calculation of the Brauer group of $BG$, for $G$ a smooth connected semisimple linear algebraic group.

The Brauer group of $BG$ and gerbe structures of moduli spaces

TL;DR

This work determines the cohomological Brauer group of the classifying stack for a smooth connected semisimple group by identifying , where is the fundamental group of and . It then analyzes Brauer-Severi stacks over and shows that the Brauer class of a stack of the form is given by , where acts on via a character . Building on this, the paper studies μ_N-gerbes arising from the inertia of the moduli stack of genus curves, focusing on components with genus-0 quotients; for these, it relates the Brauer class to a parity condition on the data describing ramification. The main results show that the Brauer group is either or depending on parity, and that the Brauer class of is the pullback , with determining triviality; when is even the class can be described via a product of simpler gerbes. The approach combines a detailed computation of Br, a stack-theoretic interpretation of Brauer-Severi varieties, rigidification techniques, and Gabber-type arguments to transfer local to global Brauer data on the moduli stacks.

Abstract

We study the -gerbe of curves of genus with an order automorphism, and explore what corresponding -cohomology classes the components of this stack can have. In particular, we look at curves whose quotients by the order automorphism are genus 0, and completely determine the Brauer classes of these gerbes. The key technical input is the calculation of the Brauer group of , for a smooth connected semisimple linear algebraic group.
Paper Structure (8 sections, 17 theorems, 112 equations)

This paper contains 8 sections, 17 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Brauer Group of $BG$
  4. Brauer-Severi stacks over $BG$
  5. The Moduli Spaces $\mathscr M_A$
  6. Picard stack and Picard scheme
  7. Rigidification of $\overline{\mathscr M_A}$
  8. Proof of Theorems \ref{['thm:N_A']} and \ref{['thm:M_A']}

Key Result

Theorem 1.1

Let $G$ be a smooth connected semisimple linear algebraic group over an algebraically closed field $k$. Let $\widetilde{G}$ be the universal cover of $G$, and $B$ the fundamental group of $G$, which fit into an exact sequence $1\to B\to \widetilde{G}\to G\to 1$. Let $X(B):=\mathrm{Hom}(B,\mathbb{G}_ For $i=2$, an isomorphism $X(B)\to H^2(BG,\mathbb{G}_m)$ is given by $(\chi:B\to \mathbb{G}_m)\maps

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • ...and 30 more