On the existence of curves with a triple point on a $K$3 surface

Abstract

Let $(S,H)$ be a general primitively polarized $K3$ surface of genus p and let $p_a(nH)$ be the arithmetic genus of $nH.$ We prove the existence in $|{\mathcal{O}}_S(nH)|$ of curves with a triple point and $A_k$-singularities. In particular, we show the existence of curves of geometric genus $g$ in $|{\mathcal{O}}_S(nH)|$ with a triple point and nodes as singularities and corresponding to regular points of their equisingular deformation locus, for every $1\le g\le p_a(nH)-3$ and (p,n)$\ne (4,1).$ Our result is obtained by studying the versal deformation space of a non-planar quadruple point.