Torsors over the Rational Double Points in Characteristic $\mathbf{p}$

Abstract

We study torsors under finite group schemes over the punctured spectrum of a singularity $x\in X$ in positive characteristic. We show that the Dieudonn\'e module of the (loc,loc)-part $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ of the local Picard sheaf can be described in terms of local Witt vector cohomology, making $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ computable. Together with the class group and the abelianised local \'etale fundamental group, $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ completely describes the finite abelian torsors over $X\setminus\{x\}$. We compute $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$ for every rational double point singularity, which complements results of Artin and Lipman, who determined ${\pi_{\mathrm{loc}}^{\mathrm{et}}}(X)$ and ${\rm Cl}(X)$. All three objects turn out to be finite. We extend the Flenner--Mumford criterion for smoothness of a normal surface germ $x \in X$ to perfect fields of positive characteristic, generalising work of Esnault and Viehweg: If $k$ is algebraically closed, then $X$ is smooth if and only if $\mathrm{Picloc}^{\mathrm{loc},\mathrm{loc}}_{X/k}$, ${\pi_{\mathrm{loc}}^{\mathrm{et}}}(X)$, and ${\rm Cl}(X)$ are trivial. Finally, we study the question whether rational double point singularities are quotient singularities by group schemes and if so, whether the group scheme is uniquely determined by the singularity. We give complete answers to both questions, except for some $D_n^r$-singularities in characteristic $2$. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.