Plus-pure thresholds of some cusp-like singularities in mixed characteristic

TL;DR

This paper investigates plus-pure thresholds of certain cusp-like singularities in mixed characteristic, demonstrating cases where these thresholds match the F-pure thresholds in positive characteristic, thus advancing understanding of singularities across characteristics.

Contribution

It establishes the relationship between plus-pure thresholds and F-pure thresholds for specific cusp-like singularities in mixed characteristic, providing new insights into their equivalence.

Findings

Plus-pure thresholds often match F-pure thresholds in studied cases.

Identifies specific singularities where thresholds agree across characteristics.

Provides examples of sporadic cases with threshold comparisons.

Abstract

Log-canonical and $F$-pure thresholds of pairs in equal characteristic admit an analog in the recent theory of singularities in mixed characteristic, which is known as the plus-pure threshold. In this paper we study plus-pure thresholds for singularities of the form $p^a + x^b \in {\bf Z}_p [[ x ]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a + x^b \in {\bf F}_p [[ t, x ]]$. We also discuss a few other sporadic examples.