Journal of the Mathematical Society of Japan

${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action

Tadashi TOMARU

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This paper presents a definition of ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb C}$. Those families are called ${\Bbb C}^*$-pencils of curves. We give the canonical method to construct them and prove some results on relations between them and normal surface singularities with ${\Bbb C}^*$-action. We also define ${\Bbb C}^*$-equivariant degeneration families of compact complex curves over ${\Bbb P}^1$. From this, it is possible to introduce a notion of dual ${\Bbb C}^*$-pencils of curves naturally. Associating it, we prove a duality for cyclic covers of normal surface singularities with ${\Bbb C}^*$-action.

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J. Math. Soc. Japan, Volume 65, Number 3 (2013), 829-885.

First available in Project Euclid: 23 July 2013

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Zentralblatt MATH identifier

Primary: 14D06: Fibrations, degenerations
Secondary: 32S25: Surface and hypersurface singularities [See also 14J17]

${\Bbb C}^*$-pencils of curves normal surface singularities with ${\Bbb C}^*$-action


TOMARU, Tadashi. ${\Bbb C}^*$-equivariant degenerations of curves and normal surface singularities with ${\Bbb C}^*$-action. J. Math. Soc. Japan 65 (2013), no. 3, 829--885. doi:10.2969/jmsj/06530829.

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  • V. I. Arnold, Local normal forms of functions, Invent. Math., 35 (1976), 87–109.
  • T. Ashikaga and M. Ishizaka, Classification of degenerations of curves of genus three via Matsumoto-Montesinos' theorem, Tohoku Math. J. (2), 54 (2002), 195–226.
  • T. Ashikaga and K. Konno, Global and Local Properties of Pencils of Algebraic Curves, In: Algebraic Geometry 2000, Adv. Stud. Pure Math., 36, Math. Sec. Japan, Tokyo, 2002, Azumino, pp.,1–49.
  • C. P. Boyer and K. Galicki, Sasakian Geometry, Oxford Math. Monogr., Oxford University Press, Oxford, 2008.
  • W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (3), 4, Springer-Verlag, Berlin, 1984.
  • M. Demazure, Anneaux gradués normaux, In: Introduction à la Théorie des Singularités, II (ed. Lê Dũng Tráng), Travaux en Cours, 37, Hermann, Paris, 1988, pp.,35–68.
  • S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom., 62 (2002), 289–349.
  • W. Ebeling and C. T. C. Wall, Kodaira singularities and an extension of Arnol'd's strange duality, Compositio Math., 56 (1985), 3–77.
  • G. Fischer, Complex Analytic Geometry, Lecture Notes in Math., 536, Springer-Verlag, Berlin, 1976.
  • A. Fujiki, On isolated singularities with ${\bm C}^*$-action, Master thesis, Kyoto University, 1973, (in Japanese).
  • A. Fujiki, On resolution of cyclic quotient singularities, Publ. Res. Inst. Math. Sci., 10 (1974/75), 293–328.
  • S. Goto and K-i. Watanabe, On graded rings. I, J. Math. Soc. Japan, 30 (1978), 179–213.
  • H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146 (1962), 311–368.
  • H. Holmann, Quotientenräume komplexer Mannigfaltigkeiten nach komplexen Lieschen Automorphismengruppen, Math. Ann., 139 (1960), 383–402.
  • H. Holmann, Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann., 150 (1963), 327–360.
  • S. Ishii, The quotients of log-canonical singularities by finite groups, In: Singularities-Sapporo 1998, Adv. Stud. Pure Math., 29, Kinokuniya, Tokyo, 2000, pp.,135–161.
  • U. Karras, On pencils of curves and deformations of minimally elliptic singularities, Math. Ann., 247 (1980), 43–65.
  • L. Kaup and B. Kaup, Holomorphic Functions of Several Variables, de Gruyter Stud. Math., 3, Walter de Gruyter & Co., Berlin, 1983.
  • Y. Kawamata, On the classification of noncomplete algebraic surfaces, In: Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., 732, Springer-Verlag, Berlin, 1979, pp.,215–232.
  • K. Kodaira, On compact analytic surface. II, Ann. of Math. (2), 77 (1963), 563–626.
  • K. Konno and D. Nagashima, Maximal ideal cycles over normal surface singularities of Brieskorn type, Osaka J. Math., 49 (2012), 225–245.
  • V. S. Kulikov, Degenerate elliptic curves and resolution of uni- and bimodal singularities, Funct. Anal. Appl., 9 (1975), 69–70.
  • J. Lu and S.-L. Tan, Inequalities between the Chern numbers of a singular fiber in a family of algebraic curves, preprint, arXiv: 1003.1767.
  • K. Matsuki, Introduction to the Mori Program, Universitext, Springer-Verlag, New York, 2002.
  • Y. Matsumoto and J. M. Montesinos-Amilibia, Pseudo-Periodic Maps and Degenerations of Riemann Surfaces, Lecture Notes in Math., 2030, Springer-Verlag, Heidelberg, 2011.
  • J. Milnor, Singular Points of Complex Hypersurfaces, Ann. Math. Stud., 61, Princeton University Press, Princeton, NJ, 1968.
  • J. Milnor and P. Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology, 9 (1970), 385–393.
  • V. Nguyen-Khac, On families of curves over ${\bm P}^1$ with small number of singular fibres, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 459–463.
  • Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscripta Math., 9 (1973), 143–186.
  • T. Okuma, The pluri-genera of surface singularities, Tohoku Math. J. (2), 50 (1998), 119–132.
  • T. Okuma, Plurigenera of Surface Singularities, Nova Science Publishers, New York, 2000.
  • P. Orlik, Seifert manifolds, Lecture Notes in Math., 291, Springer-Verlag, Berlin, New York, 1972.
  • P. Orlik and P. Wagreich, Isolated singularities of algebraic surfaces with ${\bm C}^*$-action, Ann. of Math. (2), 93 (1971), 205–228.
  • P. Orlik and P. Wagreich, Singularities of algebraic surfaces with ${\bm C}^*$-action, Math. Ann., 193 (1971), 121–135.
  • P. Orlik and P. Wagreich, Algebraic surfaces with $k^*$-action, Acta Math., 138 (1977), 43–81.
  • H. Pinkham, Normal surface singularities with ${\bm C}^*$-action, Math. Ann., 227 (1977), 183–193.
  • M. Reid, Elliptic Gorenstein singularities of surfaces, preprint, 1978.
  • O. Riemenschneider, Deformationen von Quotientensingularitäten (nach zyklischen Gruppen), Math. Ann., 209 (1974), 211–248.
  • K. Saito, Regular system of weights and associated singularities, In: Complex Analytic Singularities, Adv. Stud. Pure Math., 8, North-Holland, Amsterdam, 1987, pp.,479–526.
  • N. Saveliev, Invariants for Homology 3-Spheres, Encyclopaedia Math. Sci., Springer-Verlag, Berlin, 2002.
  • K. Stein, Analytische Zerlegungen komplexer Räume, Math. Ann., 132 (1956), 63–93.
  • J. Stevens, Elliptic surface singularities and smoothings of curves, Math. Ann., 267 (1984), 239–249.
  • M. Tomari and K-i. Watanabe, Cyclic covers of normal graded rings, Kōdai Math. J., 24 (2001), 436–457.
  • T. Tomaru, Cyclic quotients of $2$-dimensional quasi-homogeneous hypersurface singularities, Math. Z., 210 (1992), 225–244.
  • T. Tomaru, On Gorenstein surface singularities with fundamental genus $p_f \ge 2$ which satisfy some minimality conditions, Pacific J. Math., 170 (1995), 271–295.
  • T. Tomaru, On Kodaira singularities defined by $z^n=f(x,y)$, Math. Z., 236 (2001), 133–149.
  • T. Tomaru, Pinkham-Demazure construction for two dimensional cyclic quotient singularities, Tsukuba J. Math., 25 (2001), 75–83.
  • T. Tomaru, On some classes of weakly Kodaira singularities, In: Singularités Franco-Japonaises, Sémin. Congr., 10, Soc. Math. France, 2005, pp.,323–340.
  • T. Tomaru, Pencil genus for normal surface singularities, J. Math. Soc. Japan, 59 (2007), 35–80.
  • H. Tsuji, Complete negatively pinched Kähler surfaces of finite volume, Tohoku Math. J. (2), 40 (1988), 591–597.
  • K-i. Watanabe, Some remarks concerning Demazure's construction of normal graded rings, Nagoya Math. J., 83 (1981), 203–211.
  • Ki. Watanabe, On plurigenera of normal isolated singularities. I, Math. Ann., 250 (1980), 65–94.