#### Vol. 3, No. 8, 2009

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)

### Karl Schwede

Vol. 3 (2009), No. 8, 907–950
##### 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 ${\Delta }_{W}$ on $W$ satisfying $\left({K}_{X}\right){|}_{W}{\sim }_{ℚ}{K}_{W}+{\Delta }_{W}$. 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.

##### Keywords
F-pure, F-split, test ideal, log canonical, center of log canonicity, subadjunction, adjunction conjecture, different
Primary: 14B05
Secondary: 13A35