F-adjunction

Abstract

In this paper we study singularities defined by the action of Frobenius in characteristic p > 0. We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if X is a Gorenstein normal variety then to every normal center of sharp F-purity W ⊆ X such that X is F-pure at the generic point of W, there exists a canonically defined ℚ-divisor Δw on W satisfying (Kx)| w ∼ℚ Kw + Δw. Furthermore, the singularities of X near W are "the same" as the singularities of (W, Δw). As an application, we show that there are finitely many subschemes of a quasiprojective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications.