Stability of test ideals of divisors with small multiplicity

TL;DR

This paper proves the stability of test ideals at points under small multiplicity perturbations of divisors in characteristic p, and applies this to relate non-nef loci and base loci on strongly F-regular varieties.

Contribution

It establishes a stability result for test ideals with respect to small multiplicity divisors and generalizes the relation between non-nef and base loci to singular varieties in characteristic p.

Findings

Existence of a constant δ ensuring test ideal stability under small multiplicity divisors.

Non-nef locus coincides with the restricted base locus for certain divisors on strongly F-regular varieties.

Generalization of Mustață's result to singular cases in characteristic p.

Abstract

Let $(X, \Delta)$ be a log pair in characteristic $p>0$ and $P$ be a (not necessarily closed) point of $X$. We show that there exists a constant $\delta>0$ such that $\tau(X, \Delta)_P= \tau(X, \Delta + D)_P$ for each effective $\mathbb{Q}$-Cartier divisor $D$ with $\mathrm{mult}_P(D) <\delta$. As its application, we show that if $D$ is an $\mathbb{R}$-Cartier divisor on a strongly $F$-regular projective variety, then the non-nef locus of $D$ coincides with the restricted base locus of $D$. This is a generalization of a result of Musta\c{t}\v{a} to the singular case and can be viewed as a characteristic $p$ analogue of a result of Cacciola--Di Biagio.