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

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

2011-07-22

arxiv.org/abs/1107.4568v3

Mathematics

Algebraic Geometry

19 pages, some misprints corrected in Corollary 3.6, Remark 3.8 and the proof of Theorem 1.1

Scientific paper

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 $|\mathcal O_S(nH)|.$

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