Singularities of determinantal pure pairs

TL;DR

This paper investigates the singularities of determinantal affine varieties with a prime divisor, establishing conditions under which the pair exhibits purely F-regular or purely log terminal singularities, and extending results to PLT-type pairs.

Contribution

It proves that determinantal pairs are purely F-regular in characteristic p>0 and PLT in characteristic zero, also showing they are of PLT-type with suitable divisors.

Findings

Purely F-regular pairs for p>0

Purely log terminal pairs for p=0

PLT-type pairs with appropriate divisors

Abstract

Let $X$ be a generic determinantal affine variety over a perfect field of characteristic $p \geq 0$ and $P \subset X$ be a standard prime divisor generator of $\mathrm{Cl}(X) \cong \mathbb{Z}$. We prove that the pair $(X,P)$ is purely $F$-regular if $p>0$ and so that $(X,P)$ is purely log terminal (PLT) if $p=0$ and $(X,P)$ is log $\mathbb{Q}$-Gorenstein. In general, using recent results of Z. Zhuang and S. Lyu, we show that $(X,P)$ is of PLT-type, i.e. there is a $\mathbb{Q}$-divisor $\Delta$ with coefficients in $[0,1)$ such that $(X,P+\Delta)$ is PLT.