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

摘要

Let (Spec R, m) be a rational double point defined over an algebraically closed field k of characteristic p ≥ 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 (Spec R,m). We use Artin's classification of rational double points and completely determine the dimensions dim_k H_E~1(S_x) and dim_k H_E~1(S_x ⊗ O_X(E)), supplementing Wahl's theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution X → Spec R and an equisingular family that injects into a versal deformation of the rational double point (Spec R, m).