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

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

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

References in corpus (2)

• Projective Degenerations of K3 Surfaces, Gaussian Maps, and Fano Threefolds
• Rational Curves on K3 Surfaces

Cited by in corpus (1)

• Seshadri constants of $K3$ surfaces of degrees 6 and 8