Vol. 3, No. 8, 2009

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Karl Schwede

Vol. 3 (2009), No. 8, 907–950

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 X such that X is F-pure at the generic point of W, there exists a canonically defined -divisor ΔW on W satisfying (KX)|W KW + ΔW. Furthermore, the singularities of X near W are “the same” as the singularities of (W,ΔW). 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.

F-pure, F-split, test ideal, log canonical, center of log canonicity, subadjunction, adjunction conjecture, different
Mathematical Subject Classification 2000
Primary: 14B05
Secondary: 13A35
Received: 3 February 2009
Revised: 14 September 2009
Accepted: 13 October 2009
Published: 25 December 2009
Karl Schwede
Department of Mathematics
University of Michigan
East Hall
530 Church Street
Ann Arbor, MI 48109
United States