The Arithmetic of Tame Quotient Singularities in Dimension 2

Abstract

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 $\textrm {R}$. 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 $\textrm {R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\textrm {R}$, and most cyclic singularities are of type $\textrm {R}$ too.