A special case of the $\Gamma_{00}$ conjecture

TL;DR

This paper proves a special case of the $ ext{Gamma}_{00}$ conjecture, characterizing Jacobians among principally polarized abelian varieties by leveraging tangent matrix rank conditions and connections to trisecant lines on the Kummer variety.

Contribution

It establishes the $ ext{Gamma}_{00}$ conjecture for cases where the tangent matrix rank is at most 2, providing a new characterization of Jacobians within abelian varieties.

Findings

Proves the $ ext{Gamma}_{00}$ conjecture under a rank condition.

Characterizes Jacobians among abelian varieties using tangent matrix properties.

Connects the conjecture to trisecant lines on the Kummer variety.

Abstract

In this paper we prove the $\Gamma_{00}$ 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. This assumption is satisfied by Jacobians, 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 the course of his proof of the Welters' trisecant conjecture.