Theta constants associated to cubic threefolds

Authors:
Keiji Matsumoto and Tomohide Terasoma

Journal:
J. Algebraic Geom. **12** (2003), 741-775

DOI:
https://doi.org/10.1090/S1056-3911-03-00348-5

Published electronically:
June 10, 2003

MathSciNet review:
1993763

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Abstract |
References |
Additional Information

Abstract: We construct automorphic forms on the 4-dimensional complex ball which describe the inverse of a period map for marked cubic surfaces $X$ in terms of theta constants associated to the intermediate Jacobians of the triple coverings of the 3-dimensional complex projective space branching along $X$.

[A]A Atlas of finite groups, Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985.
[ACT1]ACT1 Allcock, D., Carlson, J.A. and Toledo, D., A complex hyperbolic structure for moduli space of cubic surfaces, *C.R. Acad. Sci. Paris Sér. I Math.* **326** (1998), 49–54.
[ACT2]ACT Allcock, D., Carlson, J.A. and Toledo, D., The complex hyperbolic geometry of the moduli space of cubic surfaces, *J. Algebraic Geom.* 11 (2002), 659–724.
[AF]AF Allcock, D. and Freitag, E., Cubic surfaces and Borcherds products, *Comment Math. Helv.* 77 (2002), 270–296.
[CG]CG Clemens, C.H. and Griffiths, P.A. The intermediate Jacobian of the cubic threefold, *Ann. Math. * **95** (1969), 460–541.
[C]C Coble, A, Points sets and allied Cremona transformations I, II and III, *Trans. AMS* **16** (1915), 155–198, **17** (1916), 345–385 and **18** (1917), 331–372.
[DM]DM Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, *I.H.E.S. Publ. Math.* **63** (1986), 5–89.
[DO]DO Dolgachev, I. and Ortland, D., Point sets in projective spaces and theta functions, *Asterisque.* **165** (1988).
[G]G van Geemen, B., A linear system on Naruki’s moduli space of marked cubic surfaces, *Internat. J. Math.* **13** (2002), 183–208.
[H]H Hunt, B. The geometry of some special arithmetic quotients, *LNM.* **1637**, Springer, 1996.
[I]I Igusa, J., Theta functions, Springer, 1972.
[N]N Naruki, I., Cross ratio variety as a moduli space of cubic surfaces, *Proc. London Math. Soc.* **45** (1982), 1–30.
[Ma]Ma Matsumoto, K., Theta constants associated with the cyclic triple coverings of the complex projective line branching at six points, *Publ. Res. Inst. Math. Sci.* **37** (2001), 419–440.
[Mo]Mo Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, *I.H.E.S. Publ. Math.* **63** (1986), 91–106.
[Mu]Mu Mumford, D, Prym varieties I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), 325–350, Academic Press, New York, 1974.
[MT]MT Matsumoto, K. and Terasoma, T., Comparison of the moduli spaces of cubic surfaces and certain branched coverings of the projective line, in preparation.
[P]Pic Picard, E., Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires, *Acta Math.* **2** (1883), 114–126.
[S]Shi Shiga, H., On the representation of Picard modular function by $\theta$ constants I-II, *Publ. Res. Inst. Math. Sci.* **24** (1988), 311–360.
[SY]SY Sasaki, T. and Yoshida, M. A system of differential equations in 4 variables of rank 5 invariant under the Weyl group of type $E_6$. *Kobe J. Math.* **17** (2000), 29–57.
[T]T Terada, T., Fonctions hypergéometriques $F_1$ et fonctions automorphes I, II, *J. Math. Soc. Japan* **35** (1983), 451–475; **37** (1985), 173–185.
[Y]Y Yoshida, M., A $W(E_6)$-equivariant projective embedding of the moduli space of cubic surfaces, preprint (math.AG/0002102), Kyushu University Preprint Series in Mathematics 1999-26.

[A]A Atlas of finite groups, Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A. Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray. Oxford University Press, Eynsham, 1985.
[ACT1]ACT1 Allcock, D., Carlson, J.A. and Toledo, D., A complex hyperbolic structure for moduli space of cubic surfaces, *C.R. Acad. Sci. Paris Sér. I Math.* **326** (1998), 49–54.
[ACT2]ACT Allcock, D., Carlson, J.A. and Toledo, D., The complex hyperbolic geometry of the moduli space of cubic surfaces, *J. Algebraic Geom.* 11 (2002), 659–724.
[AF]AF Allcock, D. and Freitag, E., Cubic surfaces and Borcherds products, *Comment Math. Helv.* 77 (2002), 270–296.
[CG]CG Clemens, C.H. and Griffiths, P.A. The intermediate Jacobian of the cubic threefold, *Ann. Math. * **95** (1969), 460–541.
[C]C Coble, A, Points sets and allied Cremona transformations I, II and III, *Trans. AMS* **16** (1915), 155–198, **17** (1916), 345–385 and **18** (1917), 331–372.
[DM]DM Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy, *I.H.E.S. Publ. Math.* **63** (1986), 5–89.
[DO]DO Dolgachev, I. and Ortland, D., Point sets in projective spaces and theta functions, *Asterisque.* **165** (1988).
[G]G van Geemen, B., A linear system on Naruki’s moduli space of marked cubic surfaces, *Internat. J. Math.* **13** (2002), 183–208.
[H]H Hunt, B. The geometry of some special arithmetic quotients, *LNM.* **1637**, Springer, 1996.
[I]I Igusa, J., Theta functions, Springer, 1972.
[N]N Naruki, I., Cross ratio variety as a moduli space of cubic surfaces, *Proc. London Math. Soc.* **45** (1982), 1–30.
[Ma]Ma Matsumoto, K., Theta constants associated with the cyclic triple coverings of the complex projective line branching at six points, *Publ. Res. Inst. Math. Sci.* **37** (2001), 419–440.
[Mo]Mo Mostow, G. D., Generalized Picard lattices arising from half-integral conditions, *I.H.E.S. Publ. Math.* **63** (1986), 91–106.
[Mu]Mu Mumford, D, Prym varieties I, Contributions to analysis (a collection of papers dedicated to Lipman Bers), 325–350, Academic Press, New York, 1974.
[MT]MT Matsumoto, K. and Terasoma, T., Comparison of the moduli spaces of cubic surfaces and certain branched coverings of the projective line, in preparation.
[P]Pic Picard, E., Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires, *Acta Math.* **2** (1883), 114–126.
[S]Shi Shiga, H., On the representation of Picard modular function by $\theta$ constants I-II, *Publ. Res. Inst. Math. Sci.* **24** (1988), 311–360.
[SY]SY Sasaki, T. and Yoshida, M. A system of differential equations in 4 variables of rank 5 invariant under the Weyl group of type $E_6$. *Kobe J. Math.* **17** (2000), 29–57.
[T]T Terada, T., Fonctions hypergéometriques $F_1$ et fonctions automorphes I, II, *J. Math. Soc. Japan* **35** (1983), 451–475; **37** (1985), 173–185.
[Y]Y Yoshida, M., A $W(E_6)$-equivariant projective embedding of the moduli space of cubic surfaces, preprint (math.AG/0002102), Kyushu University Preprint Series in Mathematics 1999-26.

Additional Information

**Keiji Matsumoto**

Affiliation:
Division of Mathematics, Graduate School of Science, Hokkaido University, Sapporo, Japan

Email:
matsu@math.sci.hokudai.ac.jp

**Tomohide Terasoma**

Affiliation:
Department of Mathematical Science, University of Tokyo, Komaba, Meguro, Tokyo, Japan

Email:
terasoma@ms.u-tokyo.ac.jp

Received by editor(s):
May 18, 2001

Received by editor(s) in revised form:
March 25, 2002

Published electronically:
June 10, 2003