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Limits of nodal surfaces and applications

Ciro Ciliberto, Concettina Galati

TL;DR

The paper develops a degeneration framework to study Severi varieties of nodal surfaces on smooth projective 3-folds by degenerating the ambient 3-fold to a double-normal-crossing union $A\cup B$ along $R$. It shows that the limit of a $\delta$-nodal surface whose $\delta$ nodes converge to $R$ decomposes as $S_A\cup S_B$, intersecting along a $\delta$-nodal curve, with local tacnode-type singularities $T_1$, and proves that under suitable cohomological hypotheses these $T_1$ singularities and the nodes can be smoothed independently in the total family. The authors establish a general smoothness criterion for the equisingular deformation space and connect local tacnode deformations to nodal deformations, yielding global existence results for $\delta$-nodal deformations in the general fiber. They apply the theory to produce regular components of Severi varieties for $\mathbb P^3$ and for complete intersections in $\mathbb P^4$, giving explicit bounds on $\delta$ and showing that these nodal deformations persist under degeneration. The results advance understanding of when Severi varieties of nodal surfaces are nonempty and regular, with explicit geometric constructions in classical ambient spaces.

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

Limits of nodal surfaces and applications

TL;DR

The paper develops a degeneration framework to study Severi varieties of nodal surfaces on smooth projective 3-folds by degenerating the ambient 3-fold to a double-normal-crossing union along . It shows that the limit of a -nodal surface whose nodes converge to decomposes as , intersecting along a -nodal curve, with local tacnode-type singularities , and proves that under suitable cohomological hypotheses these singularities and the nodes can be smoothed independently in the total family. The authors establish a general smoothness criterion for the equisingular deformation space and connect local tacnode deformations to nodal deformations, yielding global existence results for -nodal deformations in the general fiber. They apply the theory to produce regular components of Severi varieties for and for complete intersections in , giving explicit bounds on and showing that these nodal deformations persist under degeneration. The results advance understanding of when Severi varieties of nodal surfaces are nonempty and regular, with explicit geometric constructions in classical ambient spaces.

Abstract

Let be a flat family of projective complex 3-folds over a disc with smooth total space and smooth general fibre and whose special fiber has double normal crossing singularities, in particular, , with , smooth threefolds intersecting transversally along a smooth surface In this paper we first study the limit singularities of a --nodal surface in the general fibre , when tends to the central fibre in such a way its nodes tend to distinct points in . The result is that the limit surface is in general the union , with , smooth surfaces, intersecting on along a -nodal curve . Then we prove that, under suitable conditions, a surface as above indeed deforms to a --nodal surface in the general fibre of . As applications we prove that there are regular irreducible components of the Severi variety of degree surfaces with nodes in , for every and of the Severi variety of complete intersection -nodal surfaces of type , with in , for every
Paper Structure (17 sections, 17 theorems, 97 equations, 1 figure)

This paper contains 17 sections, 17 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Limit singularity of a node of a surface in a threefold
  3. The problem
  4. Set up
  5. Description of $\mathcal{S}_{|\mathcal{Y}_0}$ and of $S_{|\mathcal{X}_0}$
  6. Local equations of $S_0$
  7. Local deformation of a singularity of type $T_1$ to a node
  8. Deformations of surfaces with $T_1$ singularities and nodes
  9. Deformations and a smoothness criterion
  10. Deformations of $T_1$ singularities
  11. Deformations not in $\mathcal{X}_0$ of surfaces with $T_1$ singularities.
  12. Equisingular deformations of surfaces with $T_1$ singularities
  13. Global deformations of surfaces with $T_1$ singularities to nodal surfaces
  14. Applications
  15. Severi varieties
  16. ...and 2 more sections

Key Result

Theorem 2.2

Let $\mathcal{S}\subset \mathcal{Y}$ be an effective Cartier divisor as in Step 0. Then the surface $\mathcal{S}_{|\mathcal{Y}_0}$ is the union of three surfaces $\mathcal{S}_A=\mathcal{S}\cap A, \mathcal{S}_B=\mathcal{S}\cap B$ and $\mathcal{S}_{\mathcal{E}}=\mathcal{S}\cap \mathcal{E}$, where $\m

Figures (1)

  • Figure 1:

Theorems & Definitions (47)

  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 37 more