Inseparable maps on $W_n$-valued local cohomology groups of non-taut rational double point singularities and the height of K3 surfaces

TL;DR

This paper investigates non-taut rational double point singularities in certain characteristics, analyzing Frobenius actions on local cohomology groups, and relates these to the height of associated K3 surfaces.

Contribution

It computes Frobenius actions on $W_n$-valued local cohomology groups for non-taut RDPs and links the RDP isomorphism class to the height of K3 surfaces.

Findings

Frobenius acts inseparably on local cohomology groups of non-taut RDPs.

The isomorphism class of RDPs influences the height of associated K3 surfaces.

The height of K3 surfaces is determined by Frobenius action on cohomology groups.

Abstract

We consider rational double point singularities (RDPs) that are non-taut, which means that the isomorphism class is not uniquely determined from the dual graph of the minimal resolution. Such RDPs exist in characteristic $2,3,5$. We compute the actions of Frobenius, and other inseparable morphisms, on $W_n$-valued local cohomology groups of RDPs. Then we consider RDP K3 surfaces admitting non-taut RDPs. We show that the height of the K3 surface, which is also defined in terms of the Frobenius action on $W_n$-valued cohomology groups, is related to the isomorphism class of the RDP.