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A Compactification of 3 via K3 Surfaces

Published online by Cambridge University Press:  11 January 2016

Michela Artebani
Michela Artebani*
Affiliation:
Departamento de Matemática Universidad de Concepción, Casilla 160-C, Concepción, Chile, [email protected]
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Abstract

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S. Kondō defined a birational period map between the moduli space of genus three curves and a moduli space of polarized K3 surfaces. In this paper we give a resolution of the period map, providing a surjective morphism from a suitable compactification of 3 to the Baily-Borel compactification of a six dimensional ball quotient.

Keywords

14J1014J2814H10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 196 , 2009 , pp. 1 - 26
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2009

References

[1] Allcock, D., Carlson, J. A. and Toledo, D., The Moduli space of Cubic Threefolds as a Ball Quotient, to appear in Memoirs of theA.M.S., 2006.Google Scholar
[2] Baily, W. L. and Borel, A., Compactifications of arithmetic quotients of bounded symmetric domains, Ann. Math. (2), 84 (1966), 442–528.CrossRefGoogle Scholar
[3] Barth, W., Peters, C., and de Ven, A. Van, Compact complex surfaces, vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1984.Google Scholar
[4] Borel, A., Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geom., 6 (1972), 543–560.Google Scholar
[5] Brieskorn, E., Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 279–284, Gauthier-Villars, Paris, 1971.Google Scholar
[6] Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES, 63 (1986), 5–89.Google Scholar
[7] Demazure, M., Sous-groupes arithmétiques des groupes algébriques linéaires, Séminaire Bourbaki, vol. 7, pages Exp. no. 235, 209–220, Soc. Math. France, Paris, 1995.Google Scholar
[8] Dolgachev, I., Mirror symmetry for lattice polarized K3 surfaces, Algebraic Geometry 4, J. Math. Sci., 81 (1996), no. 3, 2599–2630.CrossRefGoogle Scholar
[9] Dolgachev, I. and Ortland, D., Point sets in projective spaces and theta functions, Astérisque 165, 210 pp., 1988.Google Scholar
[10] Dolgachev, I., Geemen, B. van, and Kondō, S., A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces, J. Reine Angew. Math., 588 (2005), 99–148.Google Scholar
[11] Grauert, H., Über Modifikationen und exzeptionelle analytische Mengen, Kompleks. Prostranstva (1965), 45–104; translation from Math. Ann., 146 (1962), 331–368.Google Scholar
[12] Horikawa, E., On the periods of Enriques surfaces. II, Proc. Japan Acad. Ser. Math. Sci., 53 (1977), no 2, 53–55.Google Scholar
[13] Horikawa, E., Surjectivity of the period map of K3 surfaces of degree 2, Math. Ann., 228 (1977), no. 2, 113–146.Google Scholar
[14] Hörmander, L., An introduction to complex analysis in several variables, vol. 7 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, third edition, 1990.Google Scholar
[15] Hyeon, D. and Lee, Y., Log minimal model program for the moduli space of stable curves of genus three, arXiv:math/0703093.Google Scholar
[16] Inose, H. and Shioda, T., On singular K3 surfaces, Complex Analysis and Algebraic Geometry, pp. 119–136, Iwanami Shoten, Tokyo, 1977.Google Scholar
[17] Kirwan, F., Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math., 122 (1985), 41–85.Google Scholar
[18] Kondō, S., A complex hyperbolic structure for the moduli space of curves of genus three, J. Reine Angew. Math., 525 (2000), 219–232.Google Scholar
[19] Kondō, S., The moduli space of 8 points of P1 and automorphic forms, Algebraic Geometry, pp. 89–106, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007.CrossRefGoogle Scholar
[20] Kondō, S., The moduli space of curves of genus 4 and Deligne-Mostow’s complex reflection groups, Algebraic Geometry 2000, Azumino (Hotaka), pp. 383–400, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002.Google Scholar
[21] Looijenga, E., Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J., 118 (2003), no. 1, 151–187.Google Scholar
[22] Looijenga, E., Invariants of quartic plane curves as automorphic forms, Algebraic Geometry, pp. 107–120, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007.Google Scholar
[23] Mumford, D., Stability of projective varieties, Enseignement Math. (2), 23 (1977), no. 1–2, 39–110.Google Scholar
[24] Mumford, D., Fogarty, J., and Kirwan, F., Geometric Invariant Theory, Ergeb. Math. Grenzgeb. (2), 34, Springer, 1994.Google Scholar
[25] Nikulin, V. V., Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections, J. Soviet Math., 22 (1983), 1401–1475.Google Scholar
[26] Nikulin, V. V., Integral symmetric bilinear forms and its applications, Math. USSR Izv., 14 (1980), 103–167.Google Scholar
[27] Saint-Donat, B., Projective models of K3 surfaces, American Journal of Math., 96 (1974), no. 4, 602–639.CrossRefGoogle Scholar
[28] Schubert, D., A new compactification of the moduli space of curves, Compos. Math., 78 (1991), no. 3, 297–313.Google Scholar
[29] Shah, J., Insignificant limit singularities of surfaces and their mixed Hodge structure, Ann. of Math. (2), 109 (1979), no. 3, 497–536.CrossRefGoogle Scholar
[30] Shah, J., A complete moduli space for K3 surfaces of degree 2, Ann. of Math. (2), 112 (1980), no. 3, 485–510.Google Scholar
[31] Shah, J., Degenerations of K3 surfaces of degree 4, Trans. Amer. Math. Soc., 263 (1981), no. 2, 271–308.Google Scholar
[32] Sterk, H., Compactifications of the period space of Enriques surfaces. I, Math. Z., 207 (1991), no. 1, 1–36.Google Scholar
[33] Vermeulen, A. M., Weierstrass points of weight two on curves of genus three, PhD thesis, Universiteit van Amsterdam (1983).Google Scholar
[34] Vinberg, È. B., The two most algebraic K3 surfaces, Math. Ann., 265 (1983), no. 1, 1–21.Google Scholar
[35] Yukie, A., Applications of equivariant Morse theory, PhD thesis, Harvard University (1986).Google Scholar