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

TL;DR

This paper proves the existence of curves with specific singularities on general primitively polarized K3 surfaces, including optimal cases and conditions for regularity, advancing understanding of curve singularities on K3 surfaces.

Contribution

It establishes the existence of curves with $A_k$-singularities on general K3 surfaces in various linear systems, including optimal cases, and provides regularity conditions for such families.

Findings

Existence of curves with $A_k$-singularities in $|igO_S(nH)|$

Existence of elliptic curves with cusps and nodes or tacnodes

Regularity conditions for families of curves with $A_k$-singularities

Abstract

Let $(S,H)$ be a general primitively polarized $K3$ surface. We prove the existence of curves in $|\mathcal O_S(nH)|$ 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 elliptic curves in $|\mathcal O_S(nH)|$ 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. Finally, we give a regularity condition for families of curves with only $A_k$-singularities in $|\mathcal O_S(nH)|.$