About the semiample cone of the symmetric product of a curve

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 $R(\Delta ,K):={⨁}_{(a,b)\in {\mathbb{Z}}^2}{H}^0(C^{(2)},a\Delta +bK)$, 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.

arXiv comment16 pages