F-purity of Hypersurfaces.

Abstract

Recent work of Hara and Watanabe extends the classical and much-studied notion of F-purity for rings of prime characteristic to the setting of pairs. We study the condition of F-purity for hypersurfaces (i.e., principal ideals) in F-pure rings. In particular, much of this thesis is dedicated to the study of F-pure thresholds, and we are strongly motivated by the connection between F-purity (defined via the Frobenius morphism) and log canonical singularities (defined via resolution of singularities) .

We deduce some important properties of F-purity and F-pure thresholds, and use them to show that log canonical singularities is equivalent to dense F-pure type for very general hypersurfaces in complex affine space. We also give algorithms for computing the F-pure threshold of an arbitrary diagonal or binomial hypersurface. In the diagonal case, we compute the first non-trivial test ideal of the pair, as well as some higher jumping numbers.