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On the $F$-purity of isolated log canonical singularities

Part of: General commutative ring theory Local theory Birational geometry

Published online by Cambridge University Press:  20 June 2013

Osamu Fujino  and
Shunsuke Takagi
Osamu Fujino
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan email fujino@math.kyoto-u.ac.jp
Shunsuke Takagi
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan email stakagi@ms.u-tokyo.ac.jp
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Abstract

A singularity in characteristic zero is said to be of dense $F$-pure type if its modulo $p$ reduction is locally Frobenius split for infinitely many $p$. We prove that if $x\in X$ is an isolated log canonical singularity with $\mu (x\in X)\leq 2$ (where the invariant $\mu $ is as defined in Definition 1.4), then it is of dense $F$-pure type. As a corollary, we prove the equivalence of log canonicity and being of dense $F$-pure type in the case of three-dimensional isolated $ \mathbb{Q} $-Gorenstein normal singularities.

Keywords

log canonical singularitiesF-pure singularities

MSC classification

Secondary: 14B05: Singularities 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 14E30: Minimal model program (Mori theory, extremal rays)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 9 , September 2013 , pp. 1495 - 1510
Copyright
© The Author(s) 2013 

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