Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



F-regular and F-pure rings vs. log terminal and log canonical singularities

Authors: Nobuo Hara and Kei-ichi Watanabe
Journal: J. Algebraic Geom. 11 (2002), 363-392
Published electronically: December 17, 2001
MathSciNet review: 1874118
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Abstract | References | Additional Information

Abstract: We investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to ``F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p > 0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having ``mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolution of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\mathbb Q $-divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of ``F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta )$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\mathbb Q $-divisor $\Delta $ on $\operatorname{Spec}A$. The main theorem of this paper asserts that, if $K_{A}+\Delta $ is $\mathbb Q$-Cartier, then the above three variants of F-singularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analogous to singularities of pairs in characteristic zero.

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Additional Information

Nobuo Hara
Affiliation: Department of Mathematical Sciences, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
Address at time of publication: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Kei-ichi Watanabe
Affiliation: Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakura-josui, Setagaya-ku, Tokyo 156-0045, Japan

Received by editor(s): January 17, 2000
Received by editor(s) in revised form: August 21, 2000
Published electronically: December 17, 2001
Additional Notes: Both authors are partially supported by Grant-in-Aid for Scientific Research, Japan

American Mathematical Society