The arithmetic of tame quotient singularities in dimension $2$

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.

References in corpus (4)

• An arithmetic valuative criterion for proper maps of tame algebraic stacks
• The field of moduli of varieties with a structure
• The field of moduli of plane curves
• Automorphisms and twisted forms of rings of invariants

Cited by in corpus (2)

• An arithmetic valuative criterion for proper maps of tame algebraic stacks
• The field of moduli of varieties with a structure