Fields of moduli and the arithmetic of tame quotient singularities

TL;DR

This paper extends the concept of the field of moduli to non-perfect fields, generalizes conditions for varieties to be defined over their field of moduli, and explores lifting rational points in the context of quotient singularities.

Contribution

It broadens the formalism of the field of moduli to include non-perfect fields and higher-dimensional varieties, and establishes new conditions for varieties with additional structures to be defined over their field of moduli.

Findings

Extended the definition of the field of moduli to non-perfect fields.

Proved varieties with certain automorphism properties are defined over their field of moduli.

Analyzed lifting of rational points in varieties with 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 $\gamma\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_{\gamma}$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\overline{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. Furthermore, D\`ebes 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 $\operatorname{Aut}(X,p)$ is finite, \'etale and of degree prime to $d!$ is defined over its field of moduli.