Degenerating curves and surfaces: first results

Abstract

Let $\mathcal{S} \to \mathbb{A}^1$ be a smooth family of surfaces whose general fibre is a smooth surface of ℙ³ and whose special fibre has two smooth components, intersecting transversally along a smooth curve R. We consider the Universal Severi-Enriques variety $\mathcal{V}$ on $\mathcal{S} \to \mathbb{A}^1$. The general fibre of $\mathcal{V}$ is the variety of curves on $\mathcal{S}_t$ in the linear system $|\mathcal{O}_{\mathcal{S}_t } (n)|$ with k cusps and δ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\mathcal{V}$. In this paper, we consider only the cases (k, δ) = (0, 1) and (k, δ) = (1, 0). In particular, we determine all singular curves on the special fibre of $\mathcal{S}$ which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of $\mathcal{S}$.