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On nodal deformations of singular surfaces in $\mathbb P^3$

Ciro Ciliberto, Concettina Galati

TL;DR

The paper investigates how singular degree $n$ surfaces in $\mathbb P^3$ deform to nodal surfaces within Severi varieties. It develops a degeneration framework that collapses a surface with a single ordinary or quasi-ordinary singularity to a central fibre $X_0=A\cup\Theta$ (and similarly for a double line) and uses a vanishing $h^1$ criterion to control node counts, placing the original surface in the closure of a regular Severi component. For a general ordinary multiplicity $m$, the work establishes that nearby degree $n$ surfaces sit in a regular component with $\delta={m-1\choose 2}$ nodes for small $m$ (and gives a general bound for all $m$), while for ordinary singularities along a line it proves $\delta=3n-4-\epsilon$ with $\epsilon\in\{0,1\}$ and parity constraints, allowing pinch points to deform to two nodes each. The authors also construct rational surfaces with a line of multiplicity $n-2$ to realize the required node patterns and demonstrate a genuine difference between ordinary and quasi-ordinary singularities via explicit examples, including a quasi-ordinary triple point attaining $\delta=7$ in a regular component. These results clarify how singularities can smooth to nodal configurations and provide concrete δ-boundaries and deformation mechanisms relevant to Severi varieties in projective three-space.

Abstract

In this paper we study nodal deformations of singular surfaces $S\subset \mathbb P^3$. In particular we consider the case in which $S$ has an isolated singularity of multiplicity $m$ and the case in which $S$ has only ordinary singularities along a line.

On nodal deformations of singular surfaces in $\mathbb P^3$

TL;DR

The paper investigates how singular degree surfaces in deform to nodal surfaces within Severi varieties. It develops a degeneration framework that collapses a surface with a single ordinary or quasi-ordinary singularity to a central fibre (and similarly for a double line) and uses a vanishing criterion to control node counts, placing the original surface in the closure of a regular Severi component. For a general ordinary multiplicity , the work establishes that nearby degree surfaces sit in a regular component with nodes for small (and gives a general bound for all ), while for ordinary singularities along a line it proves with and parity constraints, allowing pinch points to deform to two nodes each. The authors also construct rational surfaces with a line of multiplicity to realize the required node patterns and demonstrate a genuine difference between ordinary and quasi-ordinary singularities via explicit examples, including a quasi-ordinary triple point attaining in a regular component. These results clarify how singularities can smooth to nodal configurations and provide concrete δ-boundaries and deformation mechanisms relevant to Severi varieties in projective three-space.

Abstract

In this paper we study nodal deformations of singular surfaces . In particular we consider the case in which has an isolated singularity of multiplicity and the case in which has only ordinary singularities along a line.
Paper Structure (19 sections, 20 theorems, 63 equations, 1 figure)

This paper contains 19 sections, 20 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nodal surfaces with general plane curve section
  3. Deformations of surfaces with an ordinary singularity
  4. The deformation
  5. Nodal deformations
  6. Applications
  7. The case of general ordinary multiple point
  8. The case of a (not necessarily general) ordinary singularity
  9. An interesting example of a quasi--ordinary multiple point
  10. Some rational surfaces in $\mathbb P^3$
  11. A remark on hyperelliptic curves
  12. Certain linear systems of plane curves
  13. The image surface
  14. Nodes
  15. Deformations of surfaces with a double line
  16. ...and 4 more sections

Key Result

Proposition 2.2

Let $V$ be a good component of a Severi variety $V^{\mathbb P^3, |\mathcal{O}_{\mathbb P^3}(n)|}_\delta$, with $n\geqslant 2$. Then

Figures (1)

  • Figure 1:

Theorems & Definitions (55)

  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 45 more