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 $W\subseteq X$ such that $X$ is $F$-pure at the generic point of $W$, there exists a canonically defined $\mathbb{Q}$-divisor ${\Delta }_W$ on $W$ satisfying $\left(K_X\right){|}_W{\sim }_{\mathbb{Q}}{K}_W+{\Delta }_W$. Furthermore, the singularities of $X$ near $W$ are “the same” as the singularities of $\left(W,{\Delta }_W\right)$. 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.