Degeneration of differentials and moduli of nodal curves on 𝐾3 surfaces

Abstract

We consider, under suitable assumptions, the following situation: B is a component of the moduli space of polarized surfaces and V m,δ is the universal Severi variety over B parametrizing pairs (S, C), with (S, H) ∈ B and C ∈ |mH| irreducible with exactly δ nodes as singularities. The moduli map V → Mg of an irreducible component V of V m,δ is generically of maximal rank if and only if certain cohomology vanishings hold. Assuming there are suitable semistable degenerations of the surfaces in B, we provide sufficient conditions for the existence of an irreducible component V where these vanishings are verified. As a test, we apply this to K3 surfaces and give a new proof of a result recently independently proved by Kemeny and by the present authors.

A note on deformations of regular embeddings

“…on the general Y t . This approach has proved to be useful in the case of K3 surfaces (see [CFGK2]).…”

“…The reason we got involved in this topic, has been our work on the universal Severi variety of nodal curves on the moduli spaces of polarised K3 surfaces and on the related moduli map, cf. [CFGK1,CFGK2]. In §2 we make some considerations on this subject and explain how our results can be used to attack the study of moduli problems for Severi varieties by degeneration arguments (as we did in [CFGK2]).…”

“…[CFGK1,CFGK2]. In §2 we make some considerations on this subject and explain how our results can be used to attack the study of moduli problems for Severi varieties by degeneration arguments (as we did in [CFGK2]). We think that these ideas can be usefully applied to investigate still unexplored areas like moduli problems for Severi varieties on surfaces other than K3s, e.g., Enriques surfaces.…”