Limits of nodal surfaces and applications

Let $\mathcal{X}\to \mathbb{D}$ be a flat family of projective complex 3-folds over a disc $\mathbb{D}$ with smooth total space $\mathcal{X}$ and smooth general fibre ${\mathcal{X}}_t,$ and whose special fiber ${\mathcal{X}}_0$ has double normal crossing singularities, in particular, ${\mathcal{X}}_0=A\cup B$, with $A$, $B$ smooth threefolds intersecting transversally along a smooth surface $R=A\cap B\mathrm{.}$ In this paper we first study the limit singularities of a $\delta$–nodal surface in the general fibre $S_t\subset {\mathcal{X}}_t$, when $S_t$ tends to the central fibre in such a way its $\delta$ nodes tend to distinct points in $R$. The result is that the limit surface $S_0$ is in general the union $S_0=S_A\cup S_B$, with $S_A\subset A$, $S_B\subset B$ smooth surfaces, intersecting on $R$ along a $\delta$-nodal curve $C=S_A\cap R=S_B\cap B$. Then we prove that, under suitable conditions, a surface $S_0=S_A\cup S_B$ as above indeed deforms to a $\delta$–nodal surface in the general fibre of $\mathcal{X}\to \mathbb{D}$. As applications we prove that there are regular irreducible components of the Severi variety of degree $d$ surfaces with $\delta$ nodes in ${\mathbb{P}}^3$, for every $\delta \le \left(\genfrac{}{}{0px}{}{d-1}{2}\right)$ and of the Severi variety of complete intersection $\delta$-nodal surfaces of type $(d,h)$, with $d\ge h-1$ in ${\mathbb{P}}^4$, for every $\delta \le \left(\genfrac{}{}{0px}{}{d+3}{3}\right)-\left(\genfrac{}{}{0px}{}{d-h+1}{3}\right)-1.$