Castelnuovo theory and the geometric Schottky problem

TL;DR

This paper establishes a deep connection between Castelnuovo theory and the geometric Schottky problem, showing that certain abelian varieties satisfying Castelnuovo conditions are precisely Jacobians, with implications for the Trisecant Conjecture.

Contribution

It demonstrates that a version of the Castelnuovo Lemma characterizes Jacobians among abelian varieties, linking classical geometry to modern algebraic geometry problems.

Findings

Principally polarized abelian varieties satisfying Castelnuovo conditions are Jacobians.

A genus bound for curves in abelian varieties is provided.

The results relate Castelnuovo theory to the geometric Schottky and Trisecant Conjectures.

Abstract

We prove and conjecture results which show that Castelnuovo theory in projective space has a close analogue for abelian varieties. This is related to the geometric Schottky problem: our main result is that a principally polarized abelian variety satisfies a precise version of the Castelnuovo Lemma if and only if it is a Jacobian. This result has a surprising connection to the Trisecant Conjecture. We also give a genus bound for curves in abelian varieties.