Finite subschemes of abelian varieties and the Schottky problem (Q416016)

From the abstract: The Castelnuovo-Schottky theorem of \textit{G. Pareschi} and \textit{M. Popa} [J. Reine Angew. Math. 615, 25--44 (2008; Zbl 1142.14030)] characterizes Jacobians among indeomposable principally polarized abelian varieties \((A,\Theta)\) of dimension \(g\), by the existence of \(g+2\) points \(\Gamma \subset A\) in special position with respect to \(2\Theta\), but general with respect to \(\Theta\), and furthermore states that such a collections of points must be contained in an Abel-Jacobi curve. Building on the ideas in the original paper, we give here a self contained, scheme theoretic proof of the theorem, extending it to finite, possibly non-reduced subschemes \(\Gamma\).