1993/06/18 by Olivier Debarre, Debarre, Olivier
Mathematics · Medicine · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Berberine and alkaloids research #alg-geom #math.AG
paper · pdf · doi:10.48550/arxiv.alg-geom/9306007
11 pages, PlainTex 1.2
arxiv created 1993/06/30 · arxiv updated 2009/11/30
Let Θ be a symmetric theta divisor on an indecomposable principally polarized complex abelian variety X. The linear system |2Θ| defines a morphism K:X\ra |2Θ|^*, whose image is the Kummer variety K(X) of X. When (X,θ) is the Jacobian of an algebraic curve, there are infinitely many trisecants lines to K(X). Welters has conjectured that the existence of one trisecant line to the Kummer variety should characterize Jacobians. The purpose of this article is to show the following weak version of Welters conjecture: X is a Jacobian if and only if there exist points a,b,c of X such that (i) the subgroup of X generated by a-b and b-c is dense in X, (ii) the points K(a), K(b) and K(c) are distinct and collinear. This improves on previous results obtained by the author (Trisecant Lines And Jacobians, J. Alg. Geom. 1 (1992), 5--14). Various degenerate cases of the conjecture are also considered.