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

Abstract

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