On the Geometry of the Moduli Space of Real Binary Octics

Chu, Kenneth C. K.

Canadian journal of mathematics, Tome 63 (2011) no. 4, pp. 755-797

Résumé

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have 0, ... , 4 complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of 5-dimensional real hyperbolic space $\mathbb{R}{{\mathbb{H}}^{5}}$ by the action of an arithmetic subgroup of Isom $\left( \mathbb{R}{{\mathbb{H}}^{5}} \right)$ . These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

Détail

DOI : 10.4153/CJM-2011-026-1

Mots-clés : 32G13, 32G20, 14D05, 14D20, real binary octics, moduli space, complex hyperbolic geometry, Vinberg algorithm

Bibliographie

[1] [1] Allcock, Daniel, Carlson, James A., and Toledo, Domingo, The complex hyperbolic geometry of the moduli space of cubic surfaces. J. Algebraic Geom. 11(2002), 659–724. Google Scholar

[2] [2] Allcock, Daniel, Carlson, James A., Hyperbolic geometry and the moduli space of real binary sextics. In: Arithmetic and geometry around hypergeometric functions, Progr. Math. 260, Birkhäuser, Basel, 2007, 1–22. Google Scholar

[3] [3] Allcock, Daniel, Carlson, James A., Hyperbolic geometry and moduli of real cubic surfaces. Ann. Sci. Ec. Norm. Supér. 43(2010), 69–115. Google Scholar

[4] [4] Chung-kan Chu, Kenneth, The moduli space of real binary octics. Ph.D. dissertation, University of Utah, 2006. Google Scholar

[5] [5] Chern, S. S., Chen, W. H. and Lam, K. S., Lectures on differential geometry. World Scientific Publishing Co. Inc., River Edge, NJ, 1999. Google Scholar

[6] [6] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and 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. Google Scholar

[7] [7] Deligne, P. and Mostow, G. D., Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math. 63(1986), 5–89. Google Scholar

[8] [8] Fox, Ralph H., Covering spaces with singularities. In: A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, 243–257. Google Scholar

[9] [9] Gromov, M. and Piatetski-Shapiro, I., Nonarithmetic groups in Lobachevsky spaces. Inst. Hautes Études Sci. Publ. Math. 66(1988), 93–103. Google Scholar

[10] [10] Harris, Joe, Algebraic geometry, A First Course. Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1995. Google Scholar

[11] [11] Kondo, Shigeyuki, The moduli space of 8 points on P1 and automorphic forms. arXiv:math.AG/0504233, 2005, 1–17. Google Scholar

[12] [12] Matsumoto, Keiji and Yoshida, Masaaki, Configuration space of 8 points on the projective line and a 5-dimensional Picard modular group. Compositio Math. 86(1993), 265–280. Google Scholar

[13] [13] Miranda, Rick, Algebraic curves and Riemann surfaces. American Mathematical Society, 1995. Google Scholar

[14] [14] Picard, Émile, Sur les fonctions de deux variables indépendantes analogues aux fonctions modulaires. Acta. Math. 2(1883), 114–135. doi:10.1007/BF02612158 Google Scholar

[15] [15] Terada, Toshiaki, Fonctions hypergéométriques F1 et fonctions automorphes. I. J. Math. Soc. Japan 35(1983), 451–475. doi:10.2969/jmsj/03530451 Google Scholar

[16] [16] Terada, Toshiaki, Fonctions hypergéométriques F1 et fonctions automorphes. II. Groupes discontinus arithmétiquement définis. J. Math. Soc. Japan 37(1985), 173–185. doi:10.2969/jmsj/03720173 Google Scholar

[17] [17] Vinberg, È. B., Some arithmetical discrete groups in Lobacevskii spaces. In: Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Oxford University Press, Bombay, 1975, 323–348. Google Scholar