Fields of moduli and the arithmetic of tame quotient singularities

Abstract

Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms ${\hat{I}}^3\in \mathrm{Gal}(\overline{k}/k)$ such that the twist $X_{\hat{I}}^3$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k^{\prime }\subset \overline{k}$ such that $X$ descends to $k^{\prime }$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove 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 a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\mathrm{Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli.