Fields of moduli and the arithmetic of tame quotient singularities

Giulio Bresciani, Angelo Vistoli

Published 2022-10-10, updated 2022-10-12Version 2

Given a perfect field $k$ with algebraic closure $\bar{k}$ and a variety $X$ over $\bar{k}$, the field of moduli of $X$ is the subfield of $\bar{k}$ of elements fixed by field automorphisms $\gamma\in\operatorname{Gal}(\bar{k}/k)$ such that the twist $X_{\gamma}$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\bar{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. D\`ebes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and proved that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As an application we give conditions on the automorphism group of a variety $X$ with a smooth marked point $p$ that ensure that the pair $(X,p)$ is defined over its field of moduli.