Characterizing Jacobians via flexes of the Kummer variety

Abstract

Given an abelian variety $X$ and a point $a\in X$ we denote by $<a>$ the closure of the subgroup of $X$ generated by $a$. Let $N=2^g-1$. We denote by $\kappa :X\to \kappa (X)\subset {\mathbb{P}}^N$ 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 $a=2b\in X\setminus \{0\}$ such that $<a>$ is irreducible and $\kappa (b)$ is a flex of $\kappa (X)$.

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