Characterizing Jacobians via flexes of the Kummer variety

Abstract

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