We prove that the F-jumping numbers of the test ideal τ (X ; , a t ) are discrete and rational under the assumptions that X is a normal and F-finite scheme over a field of positive characteristic p, K X + is Q-Cartier of index not divisible p, and either X is essentially of finite type over a field or the sheaf of ideals a is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero. Mathematics Subject Classification (2000) 13A35 · 14B05M. Blickle was supported by the Deutsche Forschungsgemeinschaft (DFG) through a Heisenberg fellowship and through the SFB/Transregio 45 Periods, moduli spaces and arithmetic of algebraic varieties. K.

Let A be a d-dimensional local ring containing a field. We will prove that the highest Lyubeznik number λ d,d (A) is equal to the number of connected components of the Hochster-Huneke graph associated to B, where B = Âsh is the completion of the strict Henselization of the completion of A. This was proven by Lyubeznik in characteristic p > 0. Our statement and proof are characteristic-free.

We prove that the F -jumping coefficients of a principal ideal of an excellent regular local ring of characteristic p > 0 are all rational and form a discrete subset of R.

In this paper we show that the sets of F -jumping coefficients of ideals form discrete sets in certain graded F -finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exist for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of F -jumping coefficients of all ideals in the determinantal ring given as the quotient by 2 × 2 minors in a 2 × 3 matrix of indeterminates form discrete sets. * S, S) (cf. [S11a, §3] and [Bli09, §2]). Note that C S is an S-bimodule, as S acts on each Hom S (F α * S, S) on the left via its left action on F α * S and on the right via its right action on F α * S. When S is reduced we shall tacitly identify the inclusion S → F e * S given by s → s · 1 = s p e with the inclusion of rings S ⊂ S 1/p e . Definition 1.1. Given a ring S, an ideal a ⊆ S and a positive real number t, we define the test ideal τ (a t ) to be the unique smallest non-zero ideal J ⊆ R such that φ((a t(p e −1) J)

We extend a result by Huneke and Watanabe ([HW15]) bounding the multiplicity of F -pure local rings of prime characteristic in terms of their dimension and embedding dimensions to the case of F -injective, generalized Cohen-Macaulay rings. We then produce an upper bound for the multiplicity of any local Cohen-Macaulay ring of prime characteritic in terms of their dimensions, embedding dimensions and HSL numbers. Finally, we extend the upper bounds for the multiplicity of generalized Cohen-Macaulay rings in characteristic zero which have dense F -injective type.