Finite subschemes of abelian varieties and the Schottky problem

TL;DR

This paper provides a scheme-theoretic proof of the Castelnuovo-Schottky theorem, characterizing Jacobians among abelian varieties via special finite subschemes, and extends the theorem to include nonreduced subschemes.

Contribution

It offers a new, self-contained proof of the theorem and generalizes it to finite, possibly nonreduced subschemes.

Findings

Extended the Castelnuovo-Schottky theorem to nonreduced subschemes.

Provided a scheme-theoretic, self-contained proof of the theorem.

Characterized Jacobians using special finite subschemes in abelian varieties.

Abstract

The Castelnuovo-Schottky theorem of Pareschi-Popa characterizes Jacobians, among indecomposable principally polarized abelian varieties of dimension g, by the existence of g+2 points in general position with respect to the principal polarization, but special with respect to twice the polarization, and furthermore states that such 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 nonreduced subschemes.