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

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

2011-11-13

arxiv.org/abs/1111.3051v4

Mathematics

Algebraic Geometry

18 pages; 2 figures; references updated; typos corrected; minor changes in the abstract and in the statement of Lemma 4.1 and

Scientific paper

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.

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