In this paper we study the gonality of the normalizations of curves in the linear system $\vert H\vert$ 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 $ \vert H\vert$ 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 $\vert H\vert$ 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 $\Hilb^ k(S)$ are discussed.
Ciliberto, C., Knutsen, A. (2014). On k-gonal loci in Severi varieties on general K3 surfaces and rational curves on hyperkaehler manifolds. JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES, 101(4), 473-494 [10.1016/j.matpur.2013.06.010].
On k-gonal loci in Severi varieties on general K3 surfaces and rational curves on hyperkaehler manifolds
CILIBERTO, CIRO;
2014-01-01
Abstract
In this paper we study the gonality of the normalizations of curves in the linear system $\vert H\vert$ 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 $ \vert H\vert$ 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 $\vert H\vert$ 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 $\Hilb^ k(S)$ are discussed.File | Dimensione | Formato | |
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