Contents Online
Mathematical Research Letters
Volume 21 (2014)
Number 5
On the existence of curves with $A_k$-singularities on $K3$ surfaces
Pages: 1069 – 1109
DOI: https://dx.doi.org/10.4310/MRL.2014.v21.n5.a8
Authors
Abstract
Let $(S,H)$ be a general primitively polarized $K3$ surface. We prove the existence of irreducible curves in $\lvert \mathcal{O}_S (nH) \rvert$ with $A_k$-singularities and corresponding to regular points of the equisingular deformation locus. Our result is optimal for $n = 1$. As a corollary, we get the existence of irreducible curves in $\lvert \mathcal{O}_S (nH) \rvert$ of geometric genus $g \geq 1$ with a cusp and nodes or a simple tacnode and nodes. We obtain our result by studying the versal deformation family of the $m$-tacnode. Moreover, using results on Brill-Noether theory of curves on $K3$ surfaces, we provide a regularity condition for families of curves with only $A_k$-singularities in $\lvert \mathcal{O}_S (nH) \rvert$.
Keywords
versal deformations, tacnodes, Severi varieties, $K3$ surfaces, $A_k$-singularities
2010 Mathematics Subject Classification
14B07, 14H10, 14J28
Published 9 December 2014