Degenerating curves and surfaces: first results

Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ 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 $\delta$ 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 $(k,\delta)=(0,1)$ and $(k,\delta)=(1,0)$. 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 $\mathcal S$.