On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperk\"ahler manifolds (first version, superseded by arXiv:1204.4838)

Abstract

In this paper we study the gonality of the normalizations of curves in the linear system $|H|$ of a general primitively polarized $K3$ surface $(S,H)$ of genus $p$. We prove two main results. First we give a necessary condition on $p, g, k$ for the existence of a curve in $|H|$ with geometric genus $g$ whose normalization has a $g^ 1_k$. Secondly we prove that for $p$ even and all numerical cases compatible with the above necessary condition, there is a family of \emph{nodal} curves $|H|$ with the given $g,k$ and of dimension equal to the \emph{expected dimension} $\min\{2(k-1),g\}$. For odd $p$ the result is only slightly less sharp. 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 This version is superseded by the new submission arXiv:1204.4838 where Theorem 0.1 is improved to include the missing case and the degeneration argument in its proof is made considerably simpler. Since the degeneration argument in the present version is of a different type, and may be useful for other purposes, we choose to keep this submission as well.