The modular variety of hyperelliptic curves of genus three

Freitag, Eberhard; Manni, Riccardo

doi:10.1090/S0002-9947-2010-05024-X

The modular variety of hyperelliptic curves of genus three
HTML articles powered by AMS MathViewer

by Eberhard Freitag and Riccardo Salvati Manni PDF

Trans. Amer. Math. Soc. 363 (2011), 281-312 Request permission

Abstract:

The modular variety of nonsingular and complete hyperelliptic curves with level-two structure of genus $3$ 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 $8$ branch points. There are two important compactifications of this configuration space. The first one, $Y$, uses the semistable degenerated point configurations in $(P^1)^8$. This variety also can be identified with a Baily-Borel compactified ball-quotient $Y=\overline {\mathcal {B}/\Gamma [1-{\textrm i}]}.$ 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 $\bar M_{0,8}$ for this compactification. We have a diagram \[ \xymatrix { &\bar M_{0,8}\ar [dl]\ar [dr]&\\Y\ar @{–>}[rr]& &X\;.}\] The horizontal arrow is only birational but not everywhere regular.

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) $A,B$ such that $X = \operatorname {proj}(A)$, $Y=\operatorname {proj} (B)$.

References

Daniel Allcock and Eberhard Freitag, Cubic surfaces and Borcherds products, Comment. Math. Helv. 77 (2002), no. 2, 270–296. MR 1915042, DOI 10.1007/s00014-002-8340-4
D. Avritzer and H. Lange, The moduli spaces of hyperelliptic curves and binary forms, Math. Z. 242 (2002), no. 4, 615–632. MR 1981190, DOI 10.1007/s002090100370
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330–354. MR 422314, DOI 10.1016/S0001-8708(76)80003-5
E. Freitag, Some modular forms related to cubic surfaces, Kyungpook Math. J. 43 (2003), no. 3, 433–462. MR 2003489
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)
E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 254, Springer-Verlag, Berlin, 1983 (German). MR 871067, DOI 10.1007/978-3-642-68649-8
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, DOI 10.1090/S1056-3911-07-00460-2
J. P. Glass, Theta constants of genus three, Compositio Math. 40 (1980), no. 1, 123–137. MR 558261
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, DOI 10.1090/pspum/048/974333
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)
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 $n$ is not six, ArXiv Mathematics e-prints math.AG/0607372 (2007)
Jun-ichi Igusa, On the graded ring of theta-constants, Amer. J. Math. 86 (1964), 219–246. MR 164967, DOI 10.2307/2373041
Jun-ichi Igusa, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817–855. MR 229643, DOI 10.2307/2373243
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
Shigeyuki Kond\B{o}, The moduli space of Enriques surfaces and Borcherds products, J. Algebraic Geom. 11 (2002), no. 4, 601–627. MR 1910262, DOI 10.1090/S1056-3911-02-00301-6
Shigeyuki Kond\B{o}, The moduli space of 8 points of $\Bbb P^1$ and automorphic forms, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 89–106. MR 2296434, DOI 10.1090/conm/422/08057
Koike, K.: The projective embedding of the configuration space $X(2,8)$, Technical Reports of Mathematical Sciences, Chiba University, 16 (2000)
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
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, DOI 10.1007/978-0-8176-4578-6
Bernhard Runge, On Siegel modular forms. I, J. Reine Angew. Math. 436 (1993), 57–85. MR 1207281, DOI 10.1515/crll.1993.436.57
Bernhard Runge, On Siegel modular forms. II, Nagoya Math. J. 138 (1995), 179–197. MR 1339948, DOI 10.1017/S0027763000005237
Shigeaki Tsuyumine, Thetanullwerte on a moduli space of curves and hyperelliptic loci, Math. Z. 207 (1991), no. 4, 539–568. MR 1119956, DOI 10.1007/BF02571407