Covers of rational double points in mixed characteristic

TL;DR

This paper classifies functions leading to rational surface singularities in mixed characteristic and demonstrates the existence of regular covers for certain Gorenstein rational singularities, advancing understanding of their structure.

Contribution

It provides a classification of functions producing rational singularities in mixed characteristic and proves the existence of regular covers for specific Gorenstein rational singularities.

Findings

Classified functions for rational singularities in mixed characteristic.

Established existence of split finite covers by regular schemes.

Applied results to 2-dimensional BCM-regular singularities.

Abstract

We further the classification of rational surface singularities. Suppose $(S, \mathfrak{n}, \mathcal{k})$ is a strictly Henselian regular local ring of mixed characteristic $(0, p > 5)$. We classify functions $f$ for which $S/(f)$ has an isolated rational singularity at the maximal ideal $\mathfrak{n}$. The classification of such functions are used to show that if $(R, \mathfrak{m}, \mathcal{k})$ is an excellent, strictly Henselian, Gorenstein rational singularity of dimension $2$ and mixed characteristic $(0, p > 5)$, then there exists a split finite cover of $\mbox{Spec}(R)$ by a regular scheme. We give an application of our result to the study of $2$-dimensional BCM-regular singularities in mixed characteristic.