F-adjunction

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 $W \subseteq X$suchthat$X$is$F$-pureatthegenericpointof$W$, there exists a canonically defined $\text{bQ}$-divisor ${\mathrm{\Delta }}_W$ on $W$ satisfying $(K_X){\mathrm{\mid }}_W{\sim }_{\text{bQ}}{K}_W+{\mathrm{\Delta }}_W$. Furthermore, the singularities of $X$ near $W$ are “the same” as the singularities of $(W,{\mathrm{\Delta }}_W)$. 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.