Further evaluation of Wahl vanishing theorems for surface singularities in characteristic $p$

TL;DR

This paper refines Wahl's vanishing theorems for rational double points over fields of characteristic p, providing explicit calculations of local cohomology dimensions and constructing derivations that do not lift to resolutions.

Contribution

It explicitly determines local cohomology dimensions for rational double points in various characteristics, extending Wahl's results and constructing non-liftable derivations.

Findings

Explicit dimensions of local cohomology groups are computed.

Construction of derivations that do not lift to minimal resolutions.

Identification of non-trivial equisingular families in deformation theory.

Abstract

Let $(\mathrm{Spec }R, \frak m)$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$. 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 $(\mathrm{Spec }R, \frak m)$. We use Artin's classification of rational double points and completely determine the dimensions $\dim_k H_E^1(S_X)$, $\dim_k H_E^1(S_X\otimes \mathcal O_X(E))$, supplementing Wahl's theorems. In the proof we construct derivations concretely which do not lift to the minimal resolution $X\to \mathrm{Spec }R$, as well as non-trivial equisingular families which inject into a versal deformation of the rational double point $(\mathrm{Spec }R, \mathfrak m)$.