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 ∈ X in positive characteristic.We show that the Dieudonné module of the (loc,loc)-part Picloc loc,loc X/k of the local Picard sheaf can be described in terms of local Witt vector cohomology, making Picloc loc,loc X/k computable. Together with the class group and the abelianised local étale fundamental group, Picloc loc,loc X/kcompletely describes the finite abelian torsors over X \ {x}.We compute Picloc loc,locfor every rational double point singularity, which complements results of Artin [Ar77] and Lipman [Li69], who determined π ét loc (X) and Cl(X). All three objects turn out to be finite. We extend the Flenner-Mumford criterion for smoothness of a normal surface germ x ∈ 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 Picloc loc,loc X/k , π ét loc (X), and 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 r n -singularities in characteristic 2. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.

Finite torsors over strongly $F$-regular singularities

“…The author would like to thank Christian Liedtke who first observed the results of Claim 4.15 and Claim 4.16 and made him aware of them. These results will be treated in an upcoming work by Christian Liedtke and Gebhard Martin [LMM21b].…”

Extendability of differential forms via Cartier operators

“…(1) We cannot drop the assumption that G is linearly reductive. Indeed, an E 0 8 -singularity in characteristic p = 5, which is a quotient singularity by nonlinearly reductive group scheme α 5 , violates the logarithmic extension theorem (see [LMM21b,Table 4] and [Gra21a, Example 10.1]).…”