2021/10/07 by Christian Liedtke, Liedtke, Christian, Gebhard Martin +3
Computer Science · Mathematics · #13A35 #13A50 #14J17 #14L15 #14L30 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2110.03650
openalex publication_date 2021/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Piclocloc,locX/k of the local Picard sheaf can be described in terms of local Witt vector cohomology, making Piclocloc,locX/k computable. Together with the class group and the abelianised local étale fundamental group, Piclocloc,locX/k completely describes the finite abelian torsors over X∖\x\. We compute Piclocloc,locX/k for every rational double point singularity, which complements results of Artin and Lipman, who determined πlocet(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 ∈ 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 Piclocloc,locX/k, πlocet(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 Dnr-singularities in characteristic 2. In particular, we will give examples of (F-injective) rational double points that are not quotient singularities.