A numerical condition for a deformation of a Gorenstein surface singularity to admit a simultaneous log-canonical model
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Abstract:
Let $\pi \colon X \to T$ be a deformation of a normal Gorenstein surface singularity over the complex number field $\mathbb {C}$. We assume that $T$ is a neighborhood of the origin of $\mathbb {C}$. Then we prove that $\pi$ admits a simultaneous log-canonical model if and only if an invariant $-P_t\cdot P_t$ of each fiber $X_t$ is constant.References
- Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992. Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991; Astérisque No. 211 (1992) (1992). MR 1225842
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Shihoko Ishii, Small deformations of normal singularities, Math. Ann. 275 (1986), no. 1, 139–148. MR 849059, DOI 10.1007/BF01458588
- Shihoko Ishii, The asymptotic behavior of plurigenera for a normal isolated singularity, Math. Ann. 286 (1990), no. 4, 803–812. MR 1045403, DOI 10.1007/BF01453603
- Shihoko Ishii, Simultaneous canonical models of deformations of isolated singularities, Algebraic geometry and analytic geometry (Tokyo, 1990) ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 81–100. MR 1260940
- Shuzo Izumi, A measure of integrity for local analytic algebras, Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 719–735. MR 817161, DOI 10.2977/prims/1195178926
- Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, DOI 10.2969/aspm/01010283
- Henry B. Laufer, Weak simultaneous resolution for deformations of Gorenstein surface singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, R.I., 1983, pp. 1–29. MR 713236
- Marcel Morales, Resolution of quasihomogeneous singularities and plurigenera, Compositio Math. 64 (1987), no. 3, 311–327. MR 918415
- Noboru Nakayama, Invariance of the plurigenera of algebraic varieties under minimal model conjectures, Topology 25 (1986), no. 2, 237–251. MR 837624, DOI 10.1016/0040-9383(86)90042-X
- Tomohiro Okuma, The plurigenera of Gorenstein surface singularities, Manuscripta Math. 94 (1997), no. 2, 187–194. MR 1473895, DOI 10.1007/BF02677846
- Fumio Sakai, Anticanonical models of rational surfaces, Math. Ann. 269 (1984), no. 3, 389–410. MR 761313, DOI 10.1007/BF01450701
- Masataka Tomari and Kimio Watanabe, On $L^2$-plurigenera of not-log-canonical Gorenstein isolated singularities, Proc. Amer. Math. Soc. 109 (1990), no. 4, 931–935. MR 1021213, DOI 10.1090/S0002-9939-1990-1021213-8
- Jonathan Wahl, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), no. 3, 625–637. MR 1044058, DOI 10.1090/S0894-0347-1990-1044058-X
- Kimio Watanabe, On plurigenera of normal isolated singularities. I, Math. Ann. 250 (1980), no. 1, 65–94. MR 581632, DOI 10.1007/BF01422185
Additional Information
- Tomohiro Okuma
- Affiliation: Department of Mathematics, Gunma National College of Technology, 580 Toriba, Maebashi, Gunma 371, Japan
- MR Author ID: 619386
- Email: okuma@nat.gunma-ct.ac.jp
- Received by editor(s): August 10, 1998
- Received by editor(s) in revised form: July 15, 1999, November 4, 1999, and February 7, 2000
- Published electronically: February 15, 2001
- Communicated by: Ron Donagi
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2823-2831
- MSC (2000): Primary 14B07; Secondary 14E15, 32S30, 32S45
- DOI: https://doi.org/10.1090/S0002-9939-01-05895-6
- MathSciNet review: 1840084