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.
Categories: math.AG math.NT