The modular variety of hyperelliptic curves of genus three

Authors:
Eberhard Freitag and Riccardo Salvati Manni

Journal:
Trans. Amer. Math. Soc. **363** (2011), 281-312

MSC (2010):
Primary 11F46, 11F55

Published electronically:
August 23, 2010

MathSciNet review:
2719682

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus is a 5-dimensional quasi-projective variety which admits several standard compactifications. The first one realizes this variety as a subvariety of the Siegel modular variety of level two and genus three.

It has 36 irreducible (isomorphic) components. One of the purposes of this paper will be to describe the equations of one of these components.

Two further models use the fact that hyperelliptic curves of genus three can be obtained as coverings of a projective line with branch points. There are two important compactifications of this configuration space. The first one, , uses the semistable degenerated point configurations in . This variety also can be identified with a Baily-Borel compactified ball-quotient We will describe these results in some detail and obtain new proofs including some finer results for them. The other compactification uses the fact that families of marked projective lines can degenerate to stable marked curves of genus 0.

We use the standard notation for this compactification. We have a diagram

In this paper we find another realization of this triangle which uses the fact that there are graded algebras (closely related to algebras of modular forms) such that

**[AF]**Daniel Allcock and Eberhard Freitag,*Cubic surfaces and Borcherds products*, Comment. Math. Helv.**77**(2002), no. 2, 270–296. MR**1915042**, https://doi.org/10.1007/s00014-002-8340-4**[AL]**D. Avritzer and H. Lange,*The moduli spaces of hyperelliptic curves and binary forms*, Math. Z.**242**(2002), no. 4, 615–632. MR**1981190**, https://doi.org/10.1007/s002090100370**[Bo]**Richard E. Borcherds,*Automorphic forms with singularities on Grassmannians*, Invent. Math.**132**(1998), no. 3, 491–562. MR**1625724**, https://doi.org/10.1007/s002220050232**[DP]**C. de Concini and C. Procesi,*A characteristic free approach to invariant theory*, Advances in Math.**21**(1976), no. 3, 330–354. MR**0422314****[Fr1]**E. Freitag,*Some modular forms related to cubic surfaces*, Kyungpook Math. J.**43**(2003), no. 3, 433–462. MR**2003489****[Fr2]**Freitag, E.:*Comparison of different models of the moduli space of marked cubic surfaces,*Proceedings of Japanese-German Seminar, Ryushi-do, edited by T. Ibukyama and W. Kohnen, 74-79 (2002)**[Fr3]**E. Freitag,*Siegelsche Modulfunktionen*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR**871067****[FS]**Eberhard Freitag and Riccardo Salvati Manni,*Modular forms for the even modular lattice of signature (2,10)*, J. Algebraic Geom.**16**(2007), no. 4, 753–791. MR**2357689**, https://doi.org/10.1090/S1056-3911-07-00460-2**[Gl]**J. P. Glass,*Theta constants of genus three*, Compositio Math.**40**(1980), no. 1, 123–137. MR**558261****[Ho]**Roger Howe,*The classical groups and invariants of binary forms*, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 133–166. MR**974333**, https://doi.org/10.1090/pspum/048/974333**[HMSV1]**Howard, B.J. Millson, J. Snowden, A. Vakil, R.:*The projective invariants of ordered points on the line*, ArXiv Mathematics e-prints math.AG/0505096 (2007)**[HMSV2]**Howard, B.J. Millson, J. Snowden, A. Vakil, R.:*The moduli space of n points on the line is cut out by simple quadrics when is not six*, ArXiv Mathematics e-prints math.AG/0607372 (2007)**[Ig1]**Jun-ichi Igusa,*On the graded ring of theta-constants*, Amer. J. Math.**86**(1964), 219–246. MR**0164967****[Ig2]**Jun-ichi Igusa,*Modular forms and projective invariants*, Amer. J. Math.**89**(1967), 817–855. MR**0229643****[Ka]**M. M. Kapranov,*Chow quotients of Grassmannians. I*, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 29–110. MR**1237834****[Ko1]**Shigeyuki Kondō,*The moduli space of Enriques surfaces and Borcherds products*, J. Algebraic Geom.**11**(2002), no. 4, 601–627. MR**1910262**, https://doi.org/10.1090/S1056-3911-02-00301-6**[Ko2]**Shigeyuki Kondō,*The moduli space of 8 points of ℙ¹ and automorphic forms*, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 89–106. MR**2296434**, https://doi.org/10.1090/conm/422/08057**[Koi]**Koike, K.:*The projective embedding of the configuration space*, Technical Reports of Mathematical Sciences, Chiba University,**16**(2000)**[MY]**Keiji Matsumoto and Masaaki Yoshida,*Configuration space of 8 points on the projective line and a 5-dimensional Picard modular group*, Compositio Math.**86**(1993), no. 3, 265–280. MR**1219628****[Mu]**David Mumford,*Tata lectures on theta. II*, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2007. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura; Reprint of the 1984 original. MR**2307768****[Ru1]**Bernhard Runge,*On Siegel modular forms. I*, J. Reine Angew. Math.**436**(1993), 57–85. MR**1207281**, https://doi.org/10.1515/crll.1993.436.57**[Ru2]**Bernhard Runge,*On Siegel modular forms. II*, Nagoya Math. J.**138**(1995), 179–197. MR**1339948****[Ts]**Shigeaki Tsuyumine,*Thetanullwerte on a moduli space of curves and hyperelliptic loci*, Math. Z.**207**(1991), no. 4, 539–568. MR**1119956**, https://doi.org/10.1007/BF02571407

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
11F46,
11F55

Retrieve articles in all journals with MSC (2010): 11F46, 11F55

Additional Information

**Eberhard Freitag**

Affiliation:
Mathematisches Institut, University of Heidelberg, Im Neuenheimer Feld 288, D69120 Heidelberg, Germany

Email:
Freitag@mathi.uni-heidelberg.de

**Riccardo Salvati Manni**

Affiliation:
Dipartimento di Matematica, University La Sapienza, Piazzale Aldo Moro, 2, I-00185 Roma, Italy

Email:
salvati@mat.uniroma1.it

DOI:
https://doi.org/10.1090/S0002-9947-2010-05024-X

Received by editor(s):
December 2, 2007

Received by editor(s) in revised form:
January 29, 2009

Published electronically:
August 23, 2010

Article copyright:
© Copyright 2010
American Mathematical Society