Curves with a triple point on a $K3$ surface

Let (S, H) be a general primitively polarized K3 surface of genus p and let pa (nH) be the arithmetic genus of nH. We prove the existence in |OS (nH)| of curves with a triple point and Ak -singularities. In particular, we show the existence of curves of geometric genus g in |O_S (nH)| with a triple point and nodes as singularities and corresponding to regular points of their equisingular deformation locus, for every 1 ≤ g ≤ pa (nH) − 3 and (p, n)= (4, 1). Our result is obtained by studying the versal deformation space of a non-planar quadruple point.