Source: http://www.ams.org/journals/jag/2002-11-02/S1056-3911-01-00306-X/
Timestamp: 2019-04-23 06:05:55+00:00

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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.
[AKM] Aberbach, I., Katzman, M. and MacCrimmon, B., Weak F-regularity deforms in -Gorenstein rings, J. Algebra 204 (1998), 281-285.
[A] Alexeev, V., Classification of log-canonical surface singularities, in ``Flips and Abundance for Algebraic Threefolds--Salt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 47-58.
[E] Ein, L., Multiplier ideals, vanishing theorems and applications: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 203-219.
[EGA] Grothendieck, A. and Dieudonné, J., Éléments de Géométrie Algébrique, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
[F] Fedder, R., F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461-480.
[FW] Fedder, R. and Watanabe, K.-i., A characterization of F-regularity in terms of F-purity, in ``Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, Springer-Verlag, New York, 1989, pp. 227-245.
[Gl] Glassbrenner, D., Strong F-regularity in images of regular local rings, Proc. Amer. Math. Soc. 124 (1996), 345-353.
[Ha1] Hara, N., F-regularity and F-purity of graded rings, J. Algebra 172 (1995), 804-818.
[Ha2] -, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), 33-53.
[Ha3] -, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.
[Ha4] -, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906.
[HH1] Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116.
[HH2] -, Tight closure and strong F-regularity, Mem. Soc. Math. France 38 (1989), 119-133.
[HH3] -, F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62.
[HR] Hochster, M. and Roberts, J., The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117-172.
[Ka] Kawamata, Y., Crepant blowing-up of 3-dimensional canonical singularities and its applications to degeneration of surfaces, Ann. Math. 127 (1988), 93-163.
[KMM] Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem: in ``Algebraic Geometry, Sendai 1985," Adv. Stud. Pure Math. 10 (1987), 283-360.
[Ko] Kollár, J., Singularities of pairs: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 221-287.
[Mc] MacCrimmon, B., Weak F-regularity is strong F-regularity for rings with isolated non- -Gorenstein points, Trans. Amer. Math. Soc. (to appear).
[MR] Mehta, V. B. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985), 27-40.
[MS1] Mehta, V. B. and Srinivas, V., Normal F-pure surface singularities, J. Algebra 143 (1991), 130-143.
[MS2] -, A characterization of rational singularities, Asian J. Math. 1 (1997), 249-278.
[N] Nakayama, N., Zariski-decomposition and abundance, RIMS preprint series 1142 (1997).
[Si] Singh, A., F-regularity does not deform, Amer. J. Math. 121 (1999), 919-929.
[Sh] Shokurov V. V., -fold log flips, Izv. Russ. A. N. Ser. Mat. 56 (1992), 105-203.
[S1] Smith, K. E., F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159-180.
[S2] -, The multiplier ideal is a universal test ideal, special volume in honor of R. Hartshorne, Comm. Algebra 28 (2000), no. 12, 5915-5929.
[S3] -, Vanishing, singularities and effective bound via prime characteristic local algebra: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997).
[W1] Watanabe, K.-i., Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203-211.
[W2] -, F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341-350.
[W3] -, F-regular and F-pure rings vs. log-terminal and log-canonical singularities, (an earlier version of the present paper).
[Wi] Williams, L. J., Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721-743.
Aberbach, I., Katzman, M. and MacCrimmon, B., Weak F-regularity deforms in -Gorenstein rings, J. Algebra 204 (1998), 281-285.
Alexeev, V., Classification of log-canonical surface singularities, in ``Flips and Abundance for Algebraic Threefolds--Salt Lake City, Utah, August 1991," Asterisque No. 211, Soc. Math. France, 1992, pp. 47-58.
Ein, L., Multiplier ideals, vanishing theorems and applications: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 203-219.
Grothendieck, A. and Dieudonné, J., Éléments de Géométrie Algébrique, Chap. IV, Publ. Math. I.H.E.S. Vol. 28, 1966.
Fedder, R., F-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461-480.
Fedder, R. and Watanabe, K.-i., A characterization of F-regularity in terms of F-purity, in ``Commutative Algebra," Math. Sci. Res. Inst. Publ. Vol. 15, Springer-Verlag, New York, 1989, pp. 227-245.
Glassbrenner, D., Strong F-regularity in images of regular local rings, Proc. Amer. Math. Soc. 124 (1996), 345-353.
Hara, N., F-regularity and F-purity of graded rings, J. Algebra 172 (1995), 804-818.
-, Classification of two-dimensional F-regular and F-pure singularities, Adv. Math. 133 (1998), 33-53.
-, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981-996.
-, Geometric interpretation of tight closure and test ideals, Trans. Amer. Math. Soc. 353 (2001), 1885-1906.
Hochster, M. and Huneke, C., Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116.
-, Tight closure and strong F-regularity, Mem. Soc. Math. France 38 (1989), 119-133.
-, F-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), 1-62.
Hochster, M. and Roberts, J., The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117-172.
Kawamata, Y., Crepant blowing-up of 3-dimensional canonical singularities and its applications to degeneration of surfaces, Ann. Math. 127 (1988), 93-163.
Kawamata, Y., Matsuda, K. and Matsuki, K., Introduction to the minimal model problem: in ``Algebraic Geometry, Sendai 1985," Adv. Stud. Pure Math. 10 (1987), 283-360.
Kollár, J., Singularities of pairs: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997), 221-287.
MacCrimmon, B., Weak F-regularity is strong F-regularity for rings with isolated non- -Gorenstein points, Trans. Amer. Math. Soc. (to appear).
Mehta, V. B. and Ramanathan, A., Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. Math. 122 (1985), 27-40.
Mehta, V. B. and Srinivas, V., Normal F-pure surface singularities, J. Algebra 143 (1991), 130-143.
-, A characterization of rational singularities, Asian J. Math. 1 (1997), 249-278.
Nakayama, N., Zariski-decomposition and abundance, RIMS preprint series 1142 (1997).
Singh, A., F-regularity does not deform, Amer. J. Math. 121 (1999), 919-929.
Shokurov V. V., -fold log flips, Izv. Russ. A. N. Ser. Mat. 56 (1992), 105-203.
Smith, K. E., F-rational rings have rational singularities, Amer. J. Math. 119 (1997), 159-180.
-, The multiplier ideal is a universal test ideal, special volume in honor of R. Hartshorne, Comm. Algebra 28 (2000), no. 12, 5915-5929.
-, Vanishing, singularities and effective bound via prime characteristic local algebra: in ``Algebraic Geometry--Santa Cruz 1995," Proc. Symp. Pure Math. 62 (1997).
Watanabe, K.-i., Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J. 83 (1981), 203-211.
-, F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71 (1991), 341-350.
-, F-regular and F-pure rings vs. log-terminal and log-canonical singularities, (an earlier version of the present paper).
Williams, L. J., Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism, J. Algebra 172 (1995), 721-743.

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