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



Karl Schwede (Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA)

Kevin Tucker (Department of Mathematics, Princeton University, Princeton, NJ 08544, USA)

Wenliang Zhang (Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA)


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).

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