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About the semiample cone of the symmetric product of a curve

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 $R(Δ,K) := \bigoplus_{(a,b)\in\mathbb{Z}^2}H^0(C^{(2)},aΔ+bK)$, where $Δ$ is the image of the diagonal and $K$ is the canonical divisor. We first show that $R(Δ,K)$ 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.