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On the nef cone of symmetric products of a generic curve. (English) Zbl 1056.14042

Let \(C\) be a general curve of genus \(g \geq 1\). For any integer \(k>0\), let \(C^{(k)}\) denote its \(k\)-th symmetric product. Let \(N^1(C^{(k)})\) be the Neron-Severi group of \(C^{(k)}\) and \(\text{Nef}(C^{(k)}) \subset N^1(C^{(k)})\) the convex cone of the nef divisors. First, the author shows how to determine \(\text{Nef}(C^{(k)})\) when \(k\) is at least the gonality of \(C\), i.e. \(g \geq \lfloor g/2\rfloor +1\). The main result is its computation in the next case: \(g\) even and \(k = g/2\).

MSC:

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14C15 (Equivariant) Chow groups and rings; motives
14H99 Curves in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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