Castelnuovo Theory and the Geometric Schottky Problem

The aim of this paper is to show that Castelnuovo theory in projective space (cf. [ACGH] Ch.III §2 and [GH] Ch.4 §3) has a precise analogue for abelian varieties. This can be quite surprisingly related in a concrete way to the geometric Schottky problem, namely the problem of identifying Jacobians among all principally polarized abelian varieties (ppav’s) via geometric conditions on the polarization. The main result is that a ppav satisfies a direct analogue of the Castelnuovo Lemma if and only if it is a Jacobian. We prove or conjecture other results which show an extremely close parallel between geometry in projective space and Schottky-type projective geometry on abelian varieties.