A Special Case of the Γ00 Conjecture
In this paper we prove the Γ00 conjecture of van Geemen and van der Geer  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  in his proof of Welters’  trisecant conjecture.
Mathematics Subject Classification (2000)14H42 14H40 32G15
KeywordsJacobian abelian variety theta function Schottky problem
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- Debarre, O.: Seshadri constants of abelian varieties. The Fano Conference, 379–394, Univ. Torino, Turin, 2004.Google Scholar
- Donagi, R.: The Schottky problem, Theory of Moduli, Lecture Notes in Mathematics 1337, Springer-Verlag 1988, 84–137.Google Scholar
- Grushevsky, S., Krichever, I. Integrable discrete Schrödinger equations and a characterization of Prym varieties by a pair of quadrisecants, preprint arXiv:0705.2829Google Scholar
- 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.Google Scholar
- Krichever, I.: Characterizing Jacobians via trisecants of the Kummer Variety. Ann. Math., to appear; math.AG/0605625.Google Scholar