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

Concettina Galati

doi:10.4171/RLM/629

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

Algebraic Geometry 2012-09-05 v4

Authors: Concettina Galati

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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.

Keywords

K3 surface triple point Ak-singularities equisingular deformation versal deformation space

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@article{arxiv.1111.3051,
  title  = {On the existence of curves with a triple point on a K3 surface},
  author = {Concettina Galati},
  journal= {arXiv preprint arXiv:1111.3051},
  year   = {2012}
}

Comments

18 pages; 2 figures; references updated; typos corrected; minor changes in the abstract and in the statement of Lemma 4.1 and Theorem 1.1; a mistake in the proof of Theorem 1.1 corrected; Remark 4.4 added; statement of Corollary 4.3 corrected; final version incorporating the referees corrections. To appear on Rend. Lincei Math. Appl

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

Abstract

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