Limits of nodal surfaces and applications

TL;DR

This paper investigates the limits of nodal surfaces in degenerating families of threefolds, establishing conditions under which nodal surfaces persist in degenerations and applying results to Severi varieties in projective spaces.

Contribution

It provides new criteria for the deformation of nodal surfaces in degenerations of threefolds with normal crossing singularities, and applies these to prove the existence of components in Severi varieties.

Findings

Limit surfaces are unions of smooth components intersecting along a nodal curve.

Under certain conditions, these limit surfaces deform to nodal surfaces in the general fiber.

Existence of regular components in Severi varieties for specified degrees and numbers of nodes.

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 $\delta\leq {{d+3}\choose 3}-{{d-h+1}\choose 3}-1.$