In this paper, we prove that the set of all $F$-pure thresholds on a fixed germ of a strongly $F$-regular pair satisfies the ascending chain condition. As a corollary, we verify the ascending chain condition for the set of all $F$-pure thresholds on smooth varieties or, more generally, on varieties with tame quotient singularities, which is an affirmative answer to a conjecture given by Blickle, Mustaţǎ and Smith.

In this paper, we study the singularities of a general hyperplane section H of a three-dimensional quasi-projective variety X over an algebraically closed field of characteristic p > 0. We prove that if X has only canonical singularities, then H has only rational double points. We also prove, under the assumption that p > 5, that if X has only klt singularities, then so does H.

Abstract. Let (X, ∆) be a log pair in characteristic p > 0 and P be a (not necessarily closed) point of X. We show that there exists a constant δ > 0 such that the test ideal τ (X, ∆), a characteristic p analogue of a multiplier ideal, does not change at P under the perturbation of ∆ by any R-divisor with multiplicity less than δ. As an application, we prove that if D is an R-Cartier R-divisor on a strongly F -regular projective variety, then the non-nef locus of D coincides with the restricted base locus of D. This is a generalization of a result of Mustaţǎ to the singular case and can be viewed as a characteristic p analogue of a result of Cacciola-Di Biagio.

Given a normal Q-Gorenstein complex variety X, we prove that if one spreads it out to a normal Q-Gorenstein scheme X of mixed characteristic whose reduction X p modulo p has normal F -pure singularities for a single prime p, then X has log canonical singularities. In addition, we show its analog for log terminal singularities, without assuming that X is Q-Gorenstein, which is a generalization of a result of Ma-Schwede. We also prove that two-dimensional strongly F -regular singularities are stable under equal characteristic deformations. Our results give an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto on deformations of linearly reductive quotient singularities.

In this paper, we prove that the set of all F -pure thresholds of ideals with fixed embedding dimension satisfies the ascending chain condition. As a corollary, given an integer d, we verify the ascending chain condition for the set of all F -pure thresholds on all d-dimensional normal l.c.i. varieties. In the process of proving these results, we also show the rationality of F -pure thresholds of ideals on non-strongly F -regular pairs.