F-adjunction

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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 -divisor ΔW on W satisfying (Kₓ)|W 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 quasi-projective variety that are compatibly split by a given Frobenius splitting. We also reinterpret Fedder's criterion in this context, which has some surprising implications.