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# Mathematical Research Letters

## Volume 19 (2012)

### Number 1

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

Pages: 191 – 197

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n1.a15

#### Authors

#### Abstract

Suppose $(X, \Delta)$ is a log-$\mathbb{Q}$-Gorenstein pair.Recent work of Blickle and the first two authors gives auniform description of the multiplier ideal $J(X;\Delta)$(in characteristic zero) and the test ideal$\tau(X;\Delta)$ (in characteristic $p > 0$) via regularalterations. While in general the alteration requireddepends heavily on $\Delta$, for a fixed Cartier divisor$D$ on $X$ it is straightforward to find a singlealteration ({e.g.,} a log resolution) computing $J(X;\Delta + \lambda D)$ for all $\lambda \geq 0$. In thispaper, we show the analogous statement in positivecharacteristic: there exists a single regular alterationcomputing $\tau(X; \Delta + \lambda D)$ for all $\lambda\geq 0$. Along the way, we also prove the discreteness andrationality for the $F$-jumping numbers of $\tau(X; \Delta+\lambda D)$ for $\lambda \geq 0$ where the index of $K_X +\Delta$ is arbitrary (and may be divisible by thecharacteristic).