Mathematical Research Letters

Volume 21 (2014)

Number 5

On the existence of curves with $A_k$-singularities on $K3$ surfaces

Pages: 1069 – 1109



Concettina Galati (Dipartimento di Matematica, Università della Calabria, Arcavacata di Rende, Italy)

Andreas Leopold Knutsen (Department of Mathematics, University of Bergen, Norway)


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 $\lvert \mathcal{O}_S (nH) \rvert$.


versal deformations, tacnodes, Severi varieties, $K3$ surfaces, $A_k$-singularities

2010 Mathematics Subject Classification

14B07, 14H10, 14J28

Full Text (PDF format)