Limits of nodal surfaces and applications

Abstract

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 $R=A\cap B.$ 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 $\delta\leq {d-1\choose 2}$ and of the Severi variety of complete intersection $\delta$-nodal surfaces of type $(d,h)$, with $d\geq h-1$ in $\mathbb P^4$, for every $\delta\leq {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$