Infinitesimal Castelnuovo Theory in Abelian Varieties

TL;DR

This paper extends Castelnuovo theory to infinitesimal settings in abelian varieties, characterizing Jacobians via extremal finite schemes contained in unique Abel-Jacobi curves.

Contribution

It introduces an infinitesimal version of Castelnuovo theory for abelian varieties, providing a new characterization of Jacobians.

Findings

An irreducible principally polarized abelian variety has a finite scheme in extremal position if and only if it is a Jacobian.

Such schemes are contained in a unique Abel-Jacobi curve.

The theory links infinitesimal geometric configurations to the classical Jacobian characterization.

Abstract

The purpose of this article is to show that the Castelnuovo theory for abelian varieties, developed by G. Pareschi and M. Popa, can be infinitesimalized. More precisely, we prove that an irreducible principally polarized abelian variety has a finite scheme in extremal position, in the sense of Castelnuovo theory for abelian varieties, if, and only if, it is a Jacobian and the scheme is contained in a unique Abel-Jacobi curve.