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

DOI:
https://doi.org/10.1090/S1056-3911-01-00306-X

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 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 -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 of a normal ring of characteristic and an effective -divisor on . The main theorem of this paper asserts that, if is -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

Email:
hara@math.tohoku.ac.jp

**Kei-ichi Watanabe**

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

Email:
watanabe@math.chs.nihon-u.ac.jp

DOI:
https://doi.org/10.1090/S1056-3911-01-00306-X

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