Effective cone of the blow up of the symmetric product of a curve

TL;DR

This paper investigates the structure of the effective cone of the blow-up of the second symmetric product of a very general curve, showing it is non-polyhedral, with special focus on hyperelliptic cases and family stability.

Contribution

It proves that for very general curves, the blow-up of their symmetric product has a non-polyhedral pseudo-effective cone, extending understanding of the cone's shape in algebraic geometry.

Findings

Non-polyhedral pseudo-effective cone for the blow-up of symmetric products of very general curves.

Polyhedrality is a closed property in families of surfaces.

Special case analysis for hyperelliptic curves.

Abstract

Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective cone. The strategy is to consider first the case of hyperelliptic curves and then to show that having polyhedral pseudo-effective cone is a closed property for families of surfaces.