F-adjunction

Details F-adjunction F-adjunction

2009-01-08

arxiv.org/abs/0901.1154v4

Algebra & Number Theory, (2009) Vol. 3, No. 8, 907-950.

Mathematics

Algebraic Geometry

31 pages; to appear in Algebra and Number Theory. Typos corrected, presentation improved throughout. Section 7 subdivided into

Scientific paper

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.

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