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

Let $(\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R,\mathfrak{m})$ 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 $(\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R,\mathfrak{m})$. 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 $X\to \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R$, as well as non-trivial equisingular families which inject into a versal deformation of the rational double point $(\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}R,\mathfrak{m})$.