## Nagoya Mathematical Journal

### A compactification of $\mathcal{M}_{3}$ via K3 surfaces

Michela Artebani

#### Abstract

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 $\mathcal{M}_{3}$ to the Baily-Borel compactification of a six dimensional ball quotient.

#### Article information

Source
Nagoya Math. J., Volume 196 (2009), 1-26.

Dates
First available in Project Euclid: 15 January 2010

https://projecteuclid.org/euclid.nmj/1263564646

Mathematical Reviews number (MathSciNet)
MR2591089

Zentralblatt MATH identifier
1184.14060

#### Citation

Artebani, Michela. A compactification of $\mathcal{M}_{3}$ via K3 surfaces. Nagoya Math. J. 196 (2009), 1--26. https://projecteuclid.org/euclid.nmj/1263564646

#### References

• D. Allcock, J. A. Carlson and D. Toledo, The Moduli space of Cubic Threefolds as a Ball Quotient, to appear in Memoirs of the A.M.S., 2006.
• W. L. Baily and A. Borel, Compactifications of arithmetic quotients of bounded symmetric domains, Ann. Math. (2), 84 (1966), 442–528.
• W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, vol. 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1984.
• A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geom., 6 (1972), 543–560.
• E. Brieskorn, Singular elements of semi-simple algebraic groups, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 279–284, Gauthier-Villars, Paris, 1971.
• P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHES, 63 (1986), 5–89.
• M. Demazure, Sous-groupes arithmétiques des groupes algébriques linéaires, Séminaire Bourbaki, vol. 7, pages Exp. no. 235, 209–220, Soc. Math. France, Paris, 1995.
• I. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, Algebraic Geometry 4, J. Math. Sci., 81 (1996), no. 3, 2599–2630.
• I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque 165, 210 pp., 1988.
• I. Dolgachev, B. van Geemen, and S. Kondō, A complex ball uniformization of the moduli space of cubic surfaces via periods of $K3$ surfaces, J. Reine Angew. Math., 588 (2005), 99–148.
• H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Kompleks. Prostranstva (1965), 45–104; translation from Math. Ann., 146 (1962), 331–368.
• E. Horikawa, On the periods of Enriques surfaces. II, Proc. Japan Acad. Ser. Math. Sci., 53 (1977), no 2, 53–55.
• E. Horikawa, Surjectivity of the period map of $K3$ surfaces of degree $2$, Math. Ann., 228 (1977), no. 2, 113–146.
• L. Hörmander, An introduction to complex analysis in several variables, vol. 7 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, third edition, 1990.
• D. Hyeon and Y. Lee, Log minimal model program for the moduli space of stable curves of genus three, arXiv:math/0703093.
• H. Inose and T. Shioda, On singular $K3$ surfaces, Complex Analysis and Algebraic Geometry, pp. 119–136, Iwanami Shoten, Tokyo, 1977.
• F. Kirwan, Partial desingularisations of quotients of nonsingular varieties and their Betti numbers, Ann. of Math., 122 (1985), 41–85.
• S. Kondō, A complex hyperbolic structure for the moduli space of curves of genus three, J. Reine Angew. Math., 525 (2000), 219–232.
• S. Kondō, The moduli space of $8$ points of $\mathbb{P}^{1}$ and automorphic forms, Algebraic Geometry, pp. 89–106, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007.
• S. Kondō, 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.
• E. Looijenga, Compactifications defined by arrangements. I. The ball quotient case, Duke Math. J., 118 (2003), no. 1, 151–187.
• E. Looijenga, Invariants of quartic plane curves as automorphic forms, Algebraic Geometry, pp. 107–120, Contemp. Math., 422, Amer. Math. Soc., Providence, RI, 2007.
• D. Mumford, Stability of projective varieties, Enseignement Math. (2), 23 (1977), no. 1-2, 39–110.
• D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, Ergeb. Math. Grenzgeb. (2), 34, Springer, 1994.
• V. V. Nikulin, 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.
• V. V. Nikulin, Integral symmetric bilinear forms and its applications, Math. USSR Izv., 14 (1980), 103–167.
• B. Saint-Donat, Projective models of $K3$ surfaces, American Journal of Math., 96 (1974), no. 4, 602–639.
• D. Schubert, A new compactification of the moduli space of curves, Compos. Math., 78 (1991), no. 3, 297–313.
• J. Shah, Insignificant limit singularities of surfaces and their mixed Hodge structure, Ann. of Math. (2), 109 (1979), no. 3, 497–536.
• J. Shah, A complete moduli space for $K3$ surfaces of degree $2$, Ann. of Math. (2), 112 (1980), no. 3, 485–510.
• J. Shah, Degenerations of $K3$ surfaces of degree $4$, Trans. Amer. Math. Soc., 263 (1981), no. 2, 271–308.
• H. Sterk, Compactifications of the period space of Enriques surfaces. I, Math. Z., 207 (1991), no. 1, 1–36.
• A. M. Vermeulen, Weierstrass points of weight two on curves of genus three, PhD thesis, Universiteit van Amsterdam (1983).
• È. B. Vinberg, The two most algebraic $K3$ surfaces, Math. Ann., 265 (1983), no. 1, 1–21.
• A. Yukie, Applications of equivariant Morse theory, PhD thesis, Harvard University (1986).