  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 $W
\subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there
exists a canonically defined $\bQ$-divisor $\Delta_{W}$ on $W$ satisfying
$(K_X)|_W \sim_{\bQ} K_{W} + \Delta_{W}$. Furthermore, the singularities of $X$
near $W$ are "the same" as the singularities of $(W, \Delta_{W})$. 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.
