Let (S, H) be a general primitively polarized K3 surface. We prove the existence of irreducible curves in |O S (nH)| with A k -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 |O S (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 of Brill-Noether theory on curves of K3 surfaces, we provide a regularity condition for families of curves with only A k -singularities in |O S (nH)|.

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of these components are unirational (resp. uniruled). In particular, this applies to components of arbitrarily large genus g and φ-invariant of the polarization.

We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is related to existence of Enriques-Fano threefolds and to curves with nodal Prym-canonical model.

Abstract. We consider modular properties of nodal curves on general K3 surfaces. Let Kp be the moduli space of primitively polarized K3 surfaces (S, L) of genus p 3 and V p,m,δ → Kp be the universal Severi variety of δ-nodal irreducible curves in |mL| on (S, L) ∈ Kp. We find conditions on p, m, δ for the existence of an irreducible component V of V p,m,δ on which the moduli map ψ : V → Mg (with g = m 2 (p − 1) + 1 − δ) has generically maximal rank differential. Our results, which for any p leave only finitely many cases unsolved and are optimal for m 5 (except for very low values of p), are summarized in Theorem 1.1 in the introduction.

We prove that the locus of Prym curves (C, η) of genus g 5 for which the Prym-canonical system |ωC (η)| is base point free but the Prym-canonical map is not an embedding is irreducible and unirational of dimension 2g + 1.