F-adjunction

TL;DR

This paper explores Frobenius-induced singularities in characteristic p, establishing an inversion of adjunction principle and linking singularities of a variety to those of its subvarieties with Frobenius splitting.

Contribution

It introduces a new inversion of adjunction result for Frobenius singularities and defines a canonical divisor on centers of F-purity, connecting their singularities.

Findings

Existence of a canonical Q-divisor on centers of F-purity.

Equivalence of singularities between a variety and its F-pure centers.

Finiteness of compatibly split subschemes in a quasi-projective variety.

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 $\bQ$-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.