The arithmetic of tame quotient singularities in dimension $2$

TL;DR

This paper classifies tame quotient surface singularities of type R, showing that most non-cyclic and many cyclic singularities of this type guarantee the lifting of rational points, with implications for moduli fields.

Contribution

It provides a complete classification of tame quotient singularities of type R in dimension 2, expanding understanding of rational point lifting in algebraic geometry.

Findings

All non-cyclic tame quotient singularities in dimension 2 are of type R.

Most cyclic tame quotient singularities in dimension 2 are of type R.

The classification aids in studying fields of moduli and extends the Lang-Nishimura theorem.

Abstract

Let $k$ be a field, $X$ a variety with tame quotient singularities and $\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is singular this might be false. For certain types of singularities the rational point is guaranteed to lift, though; these are called singularities of type $\mathrm{R}$. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang-Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type $\mathrm{R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\mathrm{R}$, and most cyclic singularities are of type $\mathrm{R}$ too.