Suppose R is a Noetherian local ring with prime characteristic p>0. In this article, we show the existence of a local numerical invariant, called the F-signature, which roughly characterizes the asymptotic growth of the number of splittings of the iterates of the Frobenius endomorphism of R. This invariant was first formally defined by C. Huneke and G. Leuschke and has previously been shown to exist only in special cases. The proof of our main result is based on the development of certain uniform Hilbert-Kunz estimates of independent interest. Additionally, we analyze the behavior of the F-signature under finite ring extensions and recover explicit formulae for the F-signatures of finite quotient singularities.Comment: 19 pages. Substantially updated from a version which was circulated on a limited basis in the fall of 201

Abstract. We derive precise transformation rules for test ideals under an arbitrary finite surjective morphism π : Y → X of normal varieties in prime characteristic p > 0. Specifically, given a Q-divisor ∆X on X and any OX -linear map T : K(Y ) → K(X), we associate a Q-divisor ∆Y on Y such that T(π * τ (Y ; ∆Y )) = τ (X; ∆X ). When π is separable and T = Tr Y /X is the field trace, we have ∆Y = π * ∆X − Ramπ where Ramπ is the ramification divisor. If in addition Tr Y /X (π * OY ) = OX , we conclude π * τ (Y ; ∆Y ) ∩ K(X) = τ (X; ∆X ) and thereby recover the analogous transformation rule to multiplier ideals in characteristic zero. Our main technique is a careful study of when an OX-linear map F * OX → OX lifts to an OY -linear map F * OY → OY , and the results obtained about these liftings are of independent interest as they relate to the theory of Frobenius splittings. In particular, again assuming Tr Y /X (π * OY ) = OX , we obtain transformation results for F -pure singularities under finite maps which mirror those for log canonical singularities in characteristic zero. Finally we explore new conditions on the singularities of the ramification locus, which imply that, for a finite extension of normal domains R ⊆ S in characteristic p > 0, the trace map Tr : Frac S → Frac R sends S onto R.

Abstract. We give characterizations of test ideals and F -rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly, we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further demonstrate how these vanishing theorems may be used to extend sections.

We prove that the localétale fundamental group of a strongly F -regular singularity is finite (and likewise for theétale fundamental group of the complement of a codimension ≥ 2 set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the F -signature of the singularity gives a bound on the size of this fundamental group. To prove these results and their corollaries, we develop new transformation rules for the F -signature under finiteétale-in-codimension-one extensions. As another consequence of these transformation rules, we also obtain purity of the branch locus over rings with mild singularities (particularly if the F -signature is > 1/2). Finally, we generalize our F -signature transformation rules to the context of pairs and not-necessarilý etale-in-codimension-one extensions, obtaining an analog of another result of Xu.

Abstract. In this article, we study the jumping numbers of an ideal in the local ring at rational singularity on a complex algebraic surface. By understanding the contributions of reduced divisors on a fixed resolution, we are able to present an algorithm for finding of the jumping numbers of the ideal. This shows, in particular, how to compute the jumping numbers of a plane curve from the numerical data of its minimal resolution. In addition, the jumping numbers of the maximal ideal at the singular point in a Du Val or toric surface singularity are computed, and applications to the smooth case are explored.