Nodal Curves with General Moduli on K3 Surfaces

Abstract

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p − 2, for 2 ≤ g = p − δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ℳ _g that associates to X the isomorphism class [C] of its normalization.

Key Words:

• K3 surfaces
• Moduli spaces
• Nodal curves

2000 Mathematics Subject Classification:

• Primary 14H10, 14H51, 14J28
• Secondary 14C05, 14D15

ACKNOWLEDGMENTS

We warmly thank B. Fantechi and M. Roth for useful conversations, and C. Voisin for her comments.

F. Flamini and E. Sernesi are members of MIUR-GNSAGA at INdAM “F. Severi”. The research of A. L. Knutsen was supported by Marie Curie Intra-European Fellowship within the 6th Framework Programme. During the preparation of the article, G. Pacienza benefited from an “accueil en délégation au CNRS.”

Notes

Communicated by L. Ein.