  Let $\Theta$ be a symmetric theta divisor on an indecomposable principally
polarized complex abelian variety $X$. The linear system $|2\Theta |$ defines a
morphism $K:X\ra |2\Theta |^*$, whose image is the Kummer variety $K(X)$ of
$X$. When $(X,\theta)$ 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.
