Modular properties of nodal curves on K3 surfaces

Abstract

In this article we are going to address the following issues: (1) the first is a rigidity property for pairs (S, C) consisting of a general projective K 3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane sectionĈ → S. We prove that a non-trivial deformation of a pair (S, C) induces a non-trivial deformation of C; (2) the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K 3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes ofĈ.

The moduli of singular curves on K3 surfaces

“…then one has H 0 (B, f * (T X )) = 0 for the general [(f ∶ B → X, L)] ∈ I in every component of V n g ⊆ T n g , by [36]. Similar bounds for the case k ≥ 2 are also stated.…”

“…In the paper [36], claims are made about the generic finiteness of η ∶ V n g,k → M p(g,k)−n and the nonsurjectivity of the Wahl map for curves parametrized by the image of η. The proof of the first statement, [36,Theorem 3.1], seems flawed to us. Indeed, the statement in Step 1 that s = (s 0 , 0) is trivial, as s defines the splitting.…”

On universal Severi varieties of low genus K3 surfaces

“…In the recent paper [15], the moduli map µ g k,h has been studied also for g ≥ 13, any k and h sufficiently large with respect to g, proving that, as one may expect, µ g k,h is generically finite to its image in these cases. The remaining cases for g, h, k are very interesting and still widely open.…”

Degeneration of differentials and moduli of nodal curves on 𝐾3 surfaces

“…We will prove the desired vanishings in (1) using Proposition 2.7. If g 15, conditions (14) and (15) in Proposition 2.7 are satisfied by Proposition 3.4(i), whereas condition (16) is satisfied by the middle horizontal sequence in (23) and the vanishings of h 1 in Proposition 3.4(i).…”