About the semiample cone of the symmetric product of a curve

TL;DR

This paper investigates the finite generation of a specific ring associated with the symmetric square of a special class of algebraic curves, linking it to torsion properties of certain divisors and analyzing its behavior in moduli space.

Contribution

It establishes a criterion for finite generation of the extended canonical ring based on torsion properties and shows this criterion holds densely in the moduli space.

Findings

Finite generation occurs iff the difference of two $g_k^1$ is torsion non-trivial.

This torsion condition holds on an analytically dense subset of the moduli space.

The study connects geometric properties of curves with algebraic properties of associated rings.

Abstract

Let $C$ be a smooth curve which is complete intersection of a quadric and a degree $k>2$ surface in $\mathbb{P}^3$ and let $C^{(2)}$ be its second symmetric power. In this paper we study the finite generation of the extended canonical ring $R(\Delta,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},a\Delta+bK)$, where $\Delta$ is the image of the diagonal and $K$ is the canonical divisor. We first show that $R(\Delta,K)$ is finitely generated if and only if the difference of the two $g_k^1$ on $C$ is torsion non-trivial and then show that this holds on an analytically dense locus of the moduli space of such curves.