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

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 $g_d^r$. 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 $g_k^1$ 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.

Minor changes. Version to appear in J. Math. Pures Appl

References in corpus (2)

• Projectivity and Birational Geometry of Bridgeland moduli spaces
• Lagrangian 4-planes in holomorphic symplectic varieties of K3^[4] type

Cited by in corpus (2)

• Projectivity and Birational Geometry of Bridgeland moduli spaces
• Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles