Perfectoid pure thresholds of lifts of rational double points

TL;DR

This paper investigates the perfectoid pure threshold invariant for lifts of rational double points in mixed characteristic, revealing its rationality, chain condition, and specific behavior in characteristic 2.

Contribution

It computes perfectoid pure thresholds for lifts of rational double points and establishes their rationality and chain condition properties.

Findings

Thresholds are contained in 

Threshold set satisfies ascending chain condition

In characteristic 2, all reciprocals of positive integers occur

Abstract

We study the perfectoid pure threshold with respect to $p$, an invariant of singularities in mixed characteristic $(0,p)$ arising from perfectoid purity. In this paper, we compute perfectoid pure thresholds for lifts of rational double points. We show that the set of such thresholds is contained in $\mathbb{Q}$ and satisfies the ascending chain condition. In characteristic $2$, all reciprocals of positive integers occur, and $0$ is the unique accumulation point.