Ciliberto, Ciro; Kouvidakis, Alexis On the symmetric product of a curve with general moduli. (English) Zbl 0967.14021 Geom. Dedicata 78, No. 3, 327-343 (1999). 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 \(\geq 9\). Reviewer: A.Del Centina (Ferrara) Cited in 1 ReviewCited in 15 Documents MSC: 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14H10 Families, moduli of curves (algebraic) 14C20 Divisors, linear systems, invertible sheaves 14J40 \(n\)-folds (\(n>4\)) Keywords:second symmetric product of a curve; Nagata type problem; Néron-Severi group; cone of the effective divisors; degeneration PDFBibTeX XMLCite \textit{C. Ciliberto} and \textit{A. Kouvidakis}, Geom. Dedicata 78, No. 3, 327--343 (1999; Zbl 0967.14021) Full Text: DOI