## Heegner divisors in the moduli space of genus three curves

HTML articles powered by AMS MathViewer

- by Michela Artebani PDF
- Trans. Amer. Math. Soc.
**360**(2008), 1581-1599 Request permission

## Abstract:

S. Kondō used periods of $K3$ surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational “splitting curves”.## References

- M. Artebani. A compactification of $\mathcal M_3$ via $K3$ surfaces.
*preprint*, 2005. - W. Barth, C. Peters, and A. Van de Ven,
*Compact complex surfaces*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR**749574**, DOI 10.1007/978-3-642-96754-2 - N. Bourbaki,
*Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR**0240238** - Igor Dolgachev and David Ortland,
*Point sets in projective spaces and theta functions*, Astérisque**165**(1988), 210 pp. (1989) (English, with French summary). MR**1007155** - Ingo Hadan,
*Tangent conics at quartic surfaces and conics in quartic double solids*, Math. Nachr.**210**(2000), 127–162. MR**1738983**, DOI 10.1002/(SICI)1522-2616(200002)210:1<127::AID-MANA127>3.3.CO;2-3 - S. Kondō. The moduli space of 8 points on $\mathbb P^1$ and automorphic forms, to appear in the Proceedings of the Conference “Algebraic geometry in honor of Igor Dolgachev”.
*math.AG/0504233*. - Shigeyuki Kond\B{o},
*A complex hyperbolic structure for the moduli space of curves of genus three*, J. Reine Angew. Math.**525**(2000), 219–232. MR**1780433**, DOI 10.1515/crll.2000.069 - 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. 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. Factor groups of groups of the automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections.
*J. Soviet Math.*, 22:1401–1475, 1983. - V.V. Nikulin. Integral symmetric bilinear forms and its applications.
*Math. USSR Izv.*, 14:103–167, 1980. - Alexius Maria Vermeulen,
*Weierstrass points of weight two on curves of genus three*, Universiteit van Amsterdam, Amsterdam, 1983. Dissertation, University of Amsterdam, Amsterdam, 1983; With a Dutch summary. MR**715084**

## Additional Information

**Michela Artebani**- Affiliation: Dipartimento di Matematica, Università di Milano, via C. Saldini 50, 20133 Milano, Italia
- MR Author ID: 744997
- Email: michela.artebani@unimi.it, artebani@mat.unimi.it
- Received by editor(s): October 12, 2005
- Received by editor(s) in revised form: February 20, 2006
- Published electronically: October 22, 2007
- Additional Notes: This work was partially supported by PRIN 2003: Spazi di moduli e teoria di Lie; GNSAGA
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**360**(2008), 1581-1599 - MSC (2000): Primary 14J10, 14J28, 14H10
- DOI: https://doi.org/10.1090/S0002-9947-07-04280-8
- MathSciNet review: 2357706