Deformations of singularities and variation of GIT quotients
HTML articles powered by AMS MathViewer
- by Radu Laza PDF
- Trans. Amer. Math. Soc. 361 (2009), 2109-2161 Request permission
Abstract:
We study the deformations of the minimally elliptic surface singularity $N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$ to the study of the moduli space of pairs $(C,L)$ consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of $K3$ surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via $K3$ surfaces and for another we obtain the full deformation space of $N_{16}$. The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of $N_{16}$.References
- V. I. Arnold, V. V. Goryunov, O. V. Lyashko, and V. A. Vasil′ev, Singularity theory. I, Springer-Verlag, Berlin, 1998. Translated from the 1988 Russian original by A. Iacob; Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [Dynamical systems. VI, Encyclopaedia Math. Sci., 6, Springer, Berlin, 1993; MR1230637 (94b:58018)]. MR 1660090, DOI 10.1007/978-3-642-58009-3
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. I, Monographs in Mathematics, vol. 82, Birkhäuser Boston, Inc., Boston, MA, 1985. The classification of critical points, caustics and wave fronts; Translated from the Russian by Ian Porteous and Mark Reynolds. MR 777682, DOI 10.1007/978-1-4612-5154-5
- V. I. Arnol′d, S. M. Guseĭn-Zade, and A. N. Varchenko, Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988. Monodromy and asymptotics of integrals; Translated from the Russian by Hugh Porteous; Translation revised by the authors and James Montaldi. MR 966191, DOI 10.1007/978-1-4612-3940-6
- Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225, DOI 10.1007/978-3-642-57739-0
- Egbert Brieskorn, The unfolding of exceptional singularities, Nova Acta Leopoldina (N.F.) 52 (1981), no. 240, 65–93. Leopoldina Symposium: Singularities (Thüringen, 1978). MR 642697
- E. Brieskorn, Die Milnorgitter der exzeptionellen unimodularen Singularitäten, Bonner Mathematische Schriften [Bonn Mathematical Publications], vol. 150, Universität Bonn, Mathematisches Institut, Bonn, 1983 (German). MR 733785
- I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630. Algebraic geometry, 4. MR 1420220, DOI 10.1007/BF02362332
- Igor V. Dolgachev and Yi Hu, Variation of geometric invariant theory quotients, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 5–56. With an appendix by Nicolas Ressayre. MR 1659282
- Wolfgang Ebeling, An arithmetic characterisation of the symmetric monodromy groups of singularities, Invent. Math. 77 (1984), no. 1, 85–99. MR 751132, DOI 10.1007/BF01389136
- Robert Friedman, A new proof of the global Torelli theorem for $K3$ surfaces, Ann. of Math. (2) 120 (1984), no. 2, 237–269. MR 763907, DOI 10.2307/2006942
- Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
- Paul Hacking, Compact moduli of plane curves, Duke Math. J. 124 (2004), no. 2, 213–257. MR 2078368, DOI 10.1215/S0012-7094-04-12421-2
- Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
- Hosung Kim and Yongnam Lee, Log canonical thresholds of semistable plane curves, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 273–280. MR 2090618, DOI 10.1017/S0305004104007649
- János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- János Kollár, Karen E. Smith, and Alessio Corti, Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, vol. 92, Cambridge University Press, Cambridge, 2004. MR 2062787, DOI 10.1017/CBO9780511734991
- Eduard Looijenga, On the semi-universal deformation of a simple-elliptic hypersurface singularity. II. The discriminant, Topology 17 (1978), no. 1, 23–40. MR 492380, DOI 10.1016/0040-9383(78)90010-1
- E. Looijenga, The smoothing components of a triangle singularity. II, Math. Ann. 269 (1984), no. 3, 357–387. MR 761312, DOI 10.1007/BF01450700
- Eduard Looijenga, Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J. 118 (2003), no. 1, 151–187. MR 1978885, DOI 10.1215/S0012-7094-03-11816-5
- D. Luna, Adhérences d’orbite et invariants, Invent. Math. 29 (1975), no. 3, 231–238 (French). MR 376704, DOI 10.1007/BF01389851
- David R. Morrison, The Clemens-Schmid exact sequence and applications, Topics in transcendental algebraic geometry (Princeton, N.J., 1981/1982) Ann. of Math. Stud., vol. 106, Princeton Univ. Press, Princeton, NJ, 1984, pp. 101–119. MR 756848
- —, The geometry of $K3$ surfaces, preprint (1988), 1–72.
- Shigeru Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, vol. 81, Cambridge University Press, Cambridge, 2003. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. MR 2004218
- David Mumford, Stability of projective varieties, Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906, DOI 10.1007/978-3-642-57916-5
- V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
- Henry C. Pinkham, Deformations of algebraic varieties with $G_{m}$ action, Astérisque, No. 20, Société Mathématique de France, Paris, 1974. MR 0376672
- Henry Pinkham, Groupe de monodromie des singularités unimodulaires exceptionnelles, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 23, A1515–A1518. MR 439840
- H. C. Pinkham, Simple elliptic singularities, Del Pezzo surfaces and Cremona transformations, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 69–71. MR 0441969
- Henry Pinkham, Singularités exceptionnelles, la dualité étrange d’Arnold et les surfaces $K-3$, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 11, A615–A618. MR 429876
- Henry Pinkham, Deformations of normal surface singularities with $C^*$ action, Math. Ann. 232 (1978), no. 1, 65–84. MR 498543, DOI 10.1007/BF01420623
- Francesco Scattone, On the compactification of moduli spaces for algebraic $K3$ surfaces, Mem. Amer. Math. Soc. 70 (1987), no. 374, x+86. MR 912636, DOI 10.1090/memo/0374
- Jayant Shah, Insignificant limit singularities of surfaces and their mixed Hodge structure, Ann. of Math. (2) 109 (1979), no. 3, 497–536. MR 534760, DOI 10.2307/1971223
- Jayant Shah, A complete moduli space for $K3$ surfaces of degree $2$, Ann. of Math. (2) 112 (1980), no. 3, 485–510. MR 595204, DOI 10.2307/1971089
- Hans Sterk, Compactifications of the period space of Enriques surfaces. I, Math. Z. 207 (1991), no. 1, 1–36. MR 1106810, DOI 10.1007/BF02571372
- Michael Thaddeus, Geometric invariant theory and flips, J. Amer. Math. Soc. 9 (1996), no. 3, 691–723. MR 1333296, DOI 10.1090/S0894-0347-96-00204-4
- Tohsuke Urabe, Dynkin graphs, Gabrièlov graphs and triangle singularities, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xvii–xviii, 163–174. MR 1709351
- È. B. Vinberg, Some arithmetical discrete groups in Lobačevskiĭ spaces, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973) Oxford Univ. Press, Bombay, 1975, pp. 323–348. MR 0422505
- C. T. C. Wall, Highly singular quintic curves, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 2, 257–277. MR 1357043, DOI 10.1017/S0305004100074144
Additional Information
- Radu Laza
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 692317
- ORCID: 0000-0001-9631-1361
- Email: rlaza@umich.edu
- Received by editor(s): August 28, 2006
- Received by editor(s) in revised form: May 21, 2007
- Published electronically: November 12, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 2109-2161
- MSC (2000): Primary 14J17, 14B07, 32S25; Secondary 14L24
- DOI: https://doi.org/10.1090/S0002-9947-08-04660-6
- MathSciNet review: 2465831