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

Abstract

Suppose $(X, Δ)$ 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 $τ(X;Δ)$ (in characteristic $p > 0$) via regular alterations. While in general the alteration required depends heavily on $Δ$, 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 $λ\geq 0$. In this paper, we show the analogous statement in positive characteristic: there exists a single regular alteration computing $τ(X; Δ+ λD)$ for all $λ\geq 0$. Along the way, we also prove the discreteness and rationality for the $F$-jumping numbers of $τ(X; Δ+ λD)$ for $λ\geq 0$ where the index of $K_X + Δ$ is arbitrary (and may be divisible by the characteristic).