Abstract
In this paper we prove the Γ00 conjecture of van Geemen and van der Geer [8] under the additional assumption that the matrix of coefficients of the tangent has rank at most 2 (see Theorem 1 for a precise formulation). This assumption is satisfied by Jacobians (see proposition 1), and thus our result gives a characterization of the locus of Jacobians among all principally polarized abelian varieties.
The proof is by reduction to the (stronger version of the) characterization of Jacobians by semidegenerate trisecants, i.e., by the existence of lines tangent to the Kummer variety at one point and intersecting it in another, proven by Krichever in [16] in his proof of Welters’ [20] trisecant conjecture.
Research is supported in part by National Science Foundation under the grant DMS-05-55867
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andreotti, A.; Mayer, A. L.: On period relations for abelian integrals on algebraic curves. Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 189–238.
Beauville, A., Debarre, O., Donagi, R., van der Geer, G. Sur les fonctions thêta d’ordre deux et les singularités du diviseur thêta. C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 481–484.
Debarre, O.: Seshadri constants of abelian varieties. The Fano Conference, 379–394, Univ. Torino, Turin, 2004.
Donagi, R.: The Schottky problem, Theory of Moduli, Lecture Notes in Mathematics 1337, Springer-Verlag 1988, 84–137.
Dubrovin, B.: The Kadomcev-Petviashvili Equation and the relations between the periods of holomorphic differentials on Riemann surfaces, Math. USSR Izvestija 19 (1982), 285–296.
Fay, J.: Theta functions on Riemann surfaces. Lecture Notes in Mathematics, Vol. 352. Springer-Verlag, Berlin-New York, 1973.
Fay, J.: On the even-order vanishing of Jacobian theta functions. Duke Math. J. 51 (1984), 109–132
van Geemen, B., van der Geer, G.: Kummer varieties and the moduli spaces of abelian varieties, Amer. J. of Math. 108 (1986), 615–642.
Grushevsky, S., Krichever, I. Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants, preprint arXiv:0705.2829
Gunning, R.C.: Some curves in abelian varieties. Invent. Math. 66 (1982), 377–389.
Gunning, R.C.: Some identities for abelian integrals. Amer. J. of Math. 108 (1986), 39–74.
Izadi, E.: Fonctions thêta du second ordre sur la Jacobienne d’une courbe lisse. Math. Ann. 289 (1991), 189–202.
Izadi, E. The geometric structure of A 4, the structure of the Prym map, double solids and Γ00-divisors. J. Reine Angew. Math. 462 (1995), 93–158
Izadi, E.: Second order theta divisors on Pryms. Bull. Soc. Math. France 127 (1999), 1–23.
Krichever, I.: Integrable linear equations and the Riemann-Schottky problem. in Algebraic geometry and number theory, 497–514, Progr. Math., 253, Birkhäuser, Boston, MA, 2006.
Krichever, I.: Characterizing Jacobians via trisecants of the Kummer Variety. Ann. Math., to appear; math.AG/0605625.
Matsusaka, T.: On a characterization of a Jacobian variety. Mem. Coll. Sc. Kyoto, Ser. A 23 (1959), 1–19.
Ran, Z.: On subvarieties of abelian varieties. Invent. Math. 62 (1981), 459–479.
Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (1986), 333–382.
Welters, G.: A criterion for Jacobi varieties. Ann. Math. 120 (1984), 497–504.
Welters, G.: The surface C — C on Jacobi varieties and second order theta functions. Acta Math. 157 (1986), 1–22.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Grushevsky, S. (2010). A Special Case of the Γ00 Conjecture. In: Alonso, M.E., Arrondo, E., Mallavibarrena, R., Sols, I. (eds) Liaison, Schottky Problem and Invariant Theory. Progress in Mathematics, vol 280. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0201-3_12
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0201-3_12
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0200-6
Online ISBN: 978-3-0346-0201-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)