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
Full-text PDF

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.

References [Enhancements On Off] (What's this?)

  • [AKM] Aberbach, I., Katzman, M. and MacCrimmon, B., Weak F-regularity deforms in $\mathbb Q$-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-$\mathbb Q$-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., $3$-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.

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