Neutral representations of finite diagonalizable group schemes and fields of moduli

TL;DR

The paper introduces the concept of neutral representations for finite group schemes, providing criteria and applications to determine when certain algebraic structures are defined over their field of moduli.

Contribution

It defines neutral representations for finite group schemes and applies this to identify classes of algebraic varieties with automorphism groups over their field of moduli.

Findings

Criteria for neutrality of representations of diagonalizable group schemes

Examples of curves and varieties with automorphism groups over their field of moduli

Generalization of previous results on automorphism groups and fields of moduli

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.