Jacobians, Anti-affine groups and torsion points

TL;DR

This paper establishes criteria for when the Jacobian of a singular curve with certain singularities is anti-affine, relating it to torsion divisors and properties of the normalization, with broad conditions ensuring anti-affineness.

Contribution

It provides new criteria for the anti-affineness of Jacobians of singular curves, especially relating to torsion points and the nature of the normalization.

Findings

Jacobian of singular curves with ordinary n-point singularities can be anti-affine under specific conditions.

For curves with a single ordinary double point, a relation with torsion divisors is established.

If the normalization is a general curve of genus at least 3, the Jacobian is always anti-affine.

Abstract

We give criteria for the Jacobian of a singular curve $X$ with at most ordinary $n$-point singularities to be anti-affine. In particular, for the case of curves with single ordinary double point we exhibit a relation with torsion divisors. If the geometric genus of the singular curve is atleast 3 and the normalization is non-hyperelliptic and non-bielliptic, then except for finitely many cases the Jacobian of $X$ is anti-affine. Furthermore, if the normalization is a general curve of genus atleast 3 then the Jacobian of $X$ is always anti-affine.