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 $\mathcal S\to\mathbb A^1$\mathrm{.}Thegeneralfibreof$\mathcal V$isthevarietyofcurveson$\mathcal S_t$inthelinearsystem$|\mathcal O_{\mathcal S_t}(n)|$with$k$cuspsand$\delta$ 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 $\mathcal{S}$.