Characterizing Jacobians via flexes of the Kummer Variety

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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Published 1 January 2006