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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
MR Author ID: 216208

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