Further Evaluation of Wahl Vanishing Theorems for Surface Singularities in Characteristic p

Abstract

Let $(SpecR,\mathfrak{m})$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\ge 0$. We evaluate further the dimensions of the local cohomology groups, which were treated by Wahl in 1975 as vanishing theorem C (resp., D) under the assumption that $p$ is a very good prime (resp., good prime) with respect to $(SpecR,\mathfrak{m})$. We use Artin’s classification of rational double points and completely determine the dimensions ${dim}_k{H}_E^1\left(S_X\right)$ and ${dim}_k{H}_E^1\left(S_X\otimes {\mathcal{O}}_X\right(E\left)\right)$, supplementing Wahl’s theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution $X\to SpecR$ and an equisingular family that injects into a versal deformation of the rational double point $(SpecR,\mathfrak{m})$.