Test ideals via a single alteration and discreteness and rationality of $F$-jumping numbers

Abstract

Suppose $(X,\Delta )$ is a log-$\bQ$-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal $\mJ(X;Δ)$ (in characteristic zero) and the test ideal $\tau (X;\Delta )$ (in characteristic $p>0$) via regular alterations. While in general the alteration required depends heavily on $\Delta$, for a fixed Cartier divisor $D$ on $X$ it is straightforward to find a single alteration (e.g. a log resolution) computing $\mJ(X; Δ+ λD)$ for all $\lambda \ge 0$. In this paper, we show the analogous statement in positive characteristic: there exists a single regular alteration computing $\tau (X;\Delta +\lambda D)$ for all $\lambda \ge 0$. Along the way, we also prove the discreteness and rationality for the $F$-jumping numbers of $\tau (X;\Delta +\lambda D)$ for $\lambda \ge 0$ where the index of $K_X+\Delta$ is arbitrary (and may be divisible by the characteristic).

arXiv comment6 pages, added Remark 3.4 (explaining a further generalization of the discreteness results) and several other minor improvements. To appear in Mathematical Research Letters