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Neutral representations of finite diagonalizable group schemes and fields of moduli

Giulio Bresciani, Angelo Vistoli, Tianzhi Yang

Abstract

We introduce the notion of a neutral representation of a finite group, or finite group scheme, $G$; a representation $V$ with the property that if a gerbe $\mathcal{G}$ over a field $k$ that is a form of the classifying stack $\mathcal{B} G$ admits a vector bundle that is a form of $V$, then it is neutral, that is, $\mathcal{G}(k)$ is not empty. We give some criteria for a representation of a finite diagonalizable group scheme to be neutral. We apply this notion to give wide classes of examples of smooth curves, or varieties with a marked point, with cyclic automorphism groups, which are defined over their field of moduli, greatly generalizing some previous results.

Neutral representations of finite diagonalizable group schemes and fields of moduli

Abstract

We introduce the notion of a neutral representation of a finite group, or finite group scheme, ; a representation with the property that if a gerbe over a field that is a form of the classifying stack admits a vector bundle that is a form of , then it is neutral, that is, is not empty. We give some criteria for a representation of a finite diagonalizable group scheme to be neutral. We apply this notion to give wide classes of examples of smooth curves, or varieties with a marked point, with cyclic automorphism groups, which are defined over their field of moduli, greatly generalizing some previous results.
Paper Structure (8 sections, 17 theorems, 2 equations)

This paper contains 8 sections, 17 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. General setup, and neutral representations
  3. Neutral representations and $R$-singularities
  4. Blended eigenspaces
  5. Periods of abelian gerbes
  6. The main theorems
  7. The proofs of the theorems
  8. Geometric applications

Key Result

Theorem 1.1

Let $p$ be a prime, $G$ a cyclic group of order $p$, $V$ a finite-dimensional representation of $G$ over an algebraically closed field of characteristic different from $p$. If $\dim V - \dim V^{G}$ is not divisible by $p$, then the representation is neutral.

Theorems & Definitions (34)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Example 3.3
  • Corollary 3.4
  • proof
  • ...and 24 more