F-adjunction

Schwede, Karl

doi:10.2140/ant.2009.3.907

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F-adjunction

Karl Schwede

Vol. 3 (2009), No. 8, 907–950

DOI: 10.2140/ant.2009.3.907

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 $ℚ$ -divisor $Δ_{W}$ on $W$ satisfying $(K_{X}) |_{W} \sim_{ℚ} K_{W} + Δ_{W}$ . Furthermore, the singularities of $X$ near $W$ are “the same” as the singularities of $(W, Δ_{W})$ . 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.

Keywords

F-pure, F-split, test ideal, log canonical, center of log canonicity, subadjunction, adjunction conjecture, different

Mathematical Subject Classification 2000

Primary: 14B05

Secondary: 13A35

Milestones

Received: 3 February 2009

Revised: 14 September 2009

Accepted: 13 October 2009

Published: 25 December 2009

Authors

Karl Schwede
Department of Mathematics University of Michigan East Hall 530 Church Street Ann Arbor, MI 48109 United States
http://www.umich.edu/~kschwede

Vol. 3, No. 8, 2009