Characterizing Jacobians via flexes of the Kummer variety

Given an abelian variety X and a point a is an element of X we denote by the closure of the subgroup of X generated by a. Let N = 2(g) - 1. We denote by kappa : X --> K(X) subset of P-N the map from X to its Kummer variety. We prove that an indecomposable abelian variety X is the Jacobian of a curve if and only if there exists a point a = 2b is an element of X \ {0} such that is irreducible and kappa(b) is a flex of kappa(X).