On the symmetric product of a curve with general moduli. (English) Zbl 0967.14021

Let \(C\) be a smooth irreducible curve of genus \(g\geq 1\) and denote by \(C^{(2)}\) its second symmetric product. On \(C^{(2)}\) one can consider “natural” divisors: For \(P\in C\), define \(X_P=\{P+Q :Q\in C\}\). Let \(x\) and \(\delta\) be the classes of \(X_P\) and \(\Delta=\{2Q: Q\in C\}\) respectively. For a curve \(C\) with general moduli, it is known that the Néron-Severi group is generated by \(x\) and \(\delta\).
This paper studies the cone of the effective divisors on \(C^{(2)}\) in the \(x,\delta/2\)-plane, by considering a degeneration to a rational curve with \(g\) nodes, first given by Franchetta. In this way the problem is reduced to a “Nagata type” problem for plane curves containing preassigned singularities at general points of the plane.
In particular the authors describe the boundary of the cone of effective divisors when \(g\) is a perfect square \(\geq 9\).


14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H10 Families, moduli of curves (algebraic)
14C20 Divisors, linear systems, invertible sheaves
14J40 \(n\)-folds (\(n>4\))
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