Stability of test ideals of divisors with small multiplicity

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 Mustaţǎ to the singular case and can be viewed as a characteristic $p$ analogue of a result of Cacciola--Di Biagio.

21pages; v2; Proposition-Definition 2.19 of our previous version was not correct, see Remark 2.20 for details, references added, several typos fixed, to appear in Mathematische Zeitschrift

References in corpus (2)

• An elementary proof of Cohen-Gabber theorem in the equal characteristic $p>0$ case
• Minimizing normalized volumes of valuations