Discreteness and rationality of \(F\)-thresholds
Test ideals are positive characteristic analogues of multiplier ideals, see for example [\textit{N. Hara} and \textit{K. Yoshida}, Trans. Am. Math. Soc. 355, No. 8, 3143--3174 (2003; Zbl 1028.13003)]. Because of this relation, it is natural to ask whether properties of multiplier ideals also hold for test ideals. One of the easiest-to-prove properties of multiplier ideals is the fact that the associated jumping numbers are discrete and rational [see for example \textit{L. Ein, R. Lazarsfeld, K. E. Smith} and \textit{D. Varolin}, Duke Math. J. 123, No. 3, 469--506 (2004; Zbl 1061.14003)]. Thus it is natural to ask if the analogous \(F\)-jumping numbers of the test ideal are also discrete and rational. This is in no way obvious. This important paper gives an affirmative answer to this question in the case that the ambient space is regular and essentially of finite type over a perfect field (or even an \(F\)-finite field). The methods of the proof have are themselves both very concrete and extremely interesting. The tools developed in this paper have since spawned further work by several people including the reviewer.
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