Let (S,H) be a general primitively polarized K3 surface. We prove the existence of irreducible curves in ∣OS(nH)∣ with Ak-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 irreducible curves in ∣OS(nH)∣ of geometric genus g≥1 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. Moreover, using results on Brill-Noether theory of curves on K3 surfaces, we provide a regularity condition for families of curves with only Ak-singularities in ∣OS(nH)∣.
On the existence of curves with A_k -singularities on K3 surfaces
GALATI, CONCETTINA;
2014-01-01
Abstract
Let (S,H) be a general primitively polarized K3 surface. We prove the existence of irreducible curves in ∣OS(nH)∣ with Ak-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 irreducible curves in ∣OS(nH)∣ of geometric genus g≥1 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. Moreover, using results on Brill-Noether theory of curves on K3 surfaces, we provide a regularity condition for families of curves with only Ak-singularities in ∣OS(nH)∣.File in questo prodotto:
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