Fregular and Fpure rings vs. log terminal and log canonical singularities
Authors:
Nobuo Hara and Keiichi Watanabe
Journal:
J. Algebraic Geom. 11 (2002), 363392
DOI:
https://doi.org/10.1090/S105639110100306X
Published electronically:
December 17, 2001
MathSciNet review:
1874118
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Abstract  References  Additional Information
Abstract: We investigate the relationship of Fregular (resp. Fpure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of Fregularity and Fpurity to “Fsingularities of pairs." The notions of Fregular and Fpure 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 “Fsingularities of pairs," namely strong Fregularity, divisorial Fregularity and Fpurity 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 Fsingularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for Fsingularities 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, Shinjukuku, Tokyo 1698555, Japan
Address at time of publication:
Mathematical Institute, Tohoku University, Sendai 9808578, Japan
Email:
hara@math.tohoku.ac.jp
Keiichi Watanabe
Affiliation:
Department of Mathematics, College of Humanities and Sciences, Nihon University, Sakurajosui, Setagayaku, Tokyo 1560045, Japan
MR Author ID:
216208
Email:
watanabe@math.chs.nihonu.ac.jp
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 GrantinAid for Scientific Research, Japan