F-adjunction

In this paper we study singularities defined by the action of Frobenius in characteristic [math] . We prove results analogous to inversion of adjunction along a center of log canonicity. For example, we show that if [math] is a Gorenstein normal variety then to every normal center of sharp [math] -purity [math] such that [math] is [math] -pure at the generic point of [math] , there exists a canonically defined [math] -divisor [math] on [math] satisfying [math] . Furthermore, the singularities of [math] near [math] are “the same” as the singularities of [math] . 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.