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

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 $p^a+x^b\in {\mathbf{Z}}_p[[x]]$, showing that in a number of cases this plus-pure threshold agrees with the $F$-pure threshold of the singularity $t^a+x^b\in {\mathbf{F}}_p[[t,x]]$. We also discuss a few other sporadic examples.

v2: fixed typos, and improved exposition following referee's comments, as well as the second proof of Lemma 2.6; 16 pages, 1 table, 1 figure