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A SPECIAL CASE OF THE Γ00 CONJECTURE

By Samuel Grushevsky

Abstract

Abstract. In this paper we prove the Γ00 conjecture of van Geemen and van der Geer [7], under the additional assumption that the matrix of coefficients of the tangent has rank 1 (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 [15] in his proof of Welters ’ [19] trisecant conjecture. The Schottky problem is the question of characterizing Jacobians of algebraic curves among all ppavs — complex principally polarized abelian varieties (A, Θ). Many approaches and solutions (or weak solutions — those that only characterize the Jacobian locus up to other irreducible components) to the problem have been developed, via the singular locus of the theta divisor (Andreotti-Mayer [1]), representability of the curves of the minimal class (Matsusaka [16] and Ran [17]), the Kadomtsev-Petviashvili equation (Shiota [18]), etc. One approach that led to various characterizations of Jacobians is via the geometry of the linear system |2Θ|. The linear subsystem Γ00 ⊂ |2Θ | is defined to consist of those sections that vanish with multiplicity at least 4 at the origin. It was shown by Dubrovin [4], Fay [6], Gunning [10] that for Jacobians of Riemann surfaces all elements of |2Θ | vanish along C − C, and shown by Welters [20] that in fact for a Jacobian of a curve C the base locus Bs(Γ00) = C − C (the validity of this equality schemetheoretically was proven by Izadi in [11]). At around the same time, van Geemen and van der Geer made the followin

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