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arXiv · 1204.4838

On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds

Ciro Ciliberto·Andreas Leopold Knutsen
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Abstract

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 hyperkähler manifold $\Hilb^ k(S)$ are discussed.

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Ciro Ciliberto, Andreas Leopold Knutsen. 2013-04-26. On $k$-gonal loci in Severi varieties on general $K3$ surfaces and rational curves on hyperkähler manifolds. https://arxiv.org/abs/1204.4838

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