Mathematics – Algebraic Geometry
Scientific paper
2012-04-21
Mathematics
Algebraic Geometry
This submission supersedes arXiv:1202.2701. The main Theorem 0.1 is improved so that part (i) is optimal, and property (ii) ab
Scientific paper
In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized complex $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, r, d$ for the existence of a curve in $|H|$ with geometric genus $g$ whose normalization has a $g^ r_d$. Secondly we prove that for all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves in $|H|$ of genus $g$ carrying a $g^1_k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\}$. Relations with the Mori cone of the hyperk\"ahler manifold $\Hilb^ k(S)$ and with conjectures by Hassett-Tschinkel and by Huybrechts-Sawon are discussed. This is an improved version of the preprint arXiv:1202.2701 with stronger main result and simplified degeneration argument.
Ciliberto Ciro
Knutsen Andreas Leopold
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