The Arithmetic of Tame Quotient Singularities in Dimension 2

Let be a field, a variety with tame quotient singularities, and a resolution of singularities. Any smooth rational point lifts to by the Lang-Nishimura theorem, but if 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. 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 in dimension; in particular, we show that every non-cyclic tame quotient singularity in dimension is of type, and most cyclic singularities are of type too.