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About the semiample cone of the symmetric product of a curve. (English) Zbl 1405.14019

Summary: 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. In case the quadric is smooth, we show that \(R(\Delta,K)\) is finitely generated if and only if the difference of the two \(g_k^1\) on \(C\) is torsion and then show that this holds on an analytically dense locus of the moduli space of such curves.

MSC:

14C20 Divisors, linear systems, invertible sheaves
14H10 Families, moduli of curves (algebraic)
14H45 Special algebraic curves and curves of low genus
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