On the existence of curves with a triple point on a K3 surface

TL;DR

This paper proves the existence of curves with triple points and specific singularities on general primitively polarized K3 surfaces, expanding understanding of their geometric properties and singularity configurations.

Contribution

It establishes the existence of such curves with prescribed singularities on K3 surfaces, using deformation theory of non-planar quadruple points, a novel approach in this context.

Findings

Existence of curves with triple points and A_k-singularities on K3 surfaces.

Construction of curves with prescribed genus and singularities.

Application of versal deformation space of quadruple points.

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 $1\leq g\leq p_a(nH)-3$ and $(\p,n)\neq (4,1).$ Our result is obtained by studying the versal deformation space of a non-planar quadruple point.