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Chambers and walls in spaces of real algebraic curves of small degrees

Victor Ivanovich Zvonilov

TL;DR

The work tackles the rigid isotopy classification of real algebraic curves, focusing on nonsingular curves of bidegree $(4,3)$ on a hyperboloid and extending to singular cases. It introduces and leverages graphs of real trigonal curves on the Hirzebruch surface $\boldsymbol{\Sigma}_3$, along with dessins, skeleta, and Nagata transformations, to map chambers and walls in the space of such curves. The main contributions include a new proof of the connectedness results for 16 singular classes, a constructive description of maximally inflected trigonal curves, and a detailed wall/deformation analysis that yields a complete classification in the $(4,3)$ setting, plus an application to genus-4 curves on a quadratic cone. This methodology links real plane curves, curves on quadrics, and trigonal curves on Hirzebruch surfaces, providing a robust framework for Hilbert’s 16th problem in small-degree cases and enhancing computational/adaptive classification via dessins and skeletons.

Abstract

This paper reviews known results on the rigid isotopy classification of plane curves of degree $m\leq6$ and curves of small degrees on quadrics. The paper's study completes the rigid isotopy classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, begun by the author in earlier works. There are given previously missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces. Adjacency graphs of chambers and walls in the spaces of these curves are presented.

Chambers and walls in spaces of real algebraic curves of small degrees

TL;DR

The work tackles the rigid isotopy classification of real algebraic curves, focusing on nonsingular curves of bidegree on a hyperboloid and extending to singular cases. It introduces and leverages graphs of real trigonal curves on the Hirzebruch surface , along with dessins, skeleta, and Nagata transformations, to map chambers and walls in the space of such curves. The main contributions include a new proof of the connectedness results for 16 singular classes, a constructive description of maximally inflected trigonal curves, and a detailed wall/deformation analysis that yields a complete classification in the setting, plus an application to genus-4 curves on a quadratic cone. This methodology links real plane curves, curves on quadrics, and trigonal curves on Hirzebruch surfaces, providing a robust framework for Hilbert’s 16th problem in small-degree cases and enhancing computational/adaptive classification via dessins and skeletons.

Abstract

This paper reviews known results on the rigid isotopy classification of plane curves of degree and curves of small degrees on quadrics. The paper's study completes the rigid isotopy classification of nonsingular real algebraic curves of bidegree (4,3) on a hyperboloid, begun by the author in earlier works. There are given previously missing proofs of the uniqueness of the connected components for 16 classes of real algebraic curves of bidegree (4,3) having a single node or a cusp. The main technical tools are graphs of real trigonal curves on Hirzebruch surfaces. Adjacency graphs of chambers and walls in the spaces of these curves are presented.

Paper Structure

This paper contains 33 sections, 15 theorems, 5 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Principal results
  3. Contents of the paper
  4. Definitions and notation
  5. Real structure
  6. Real and complex schemes
  7. Real quadrics
  8. Coding a real scheme
  9. Trigonal curves on Hirzebruch surfaces
  10. Deformations
  11. Trinomial and tetragonal curves
  12. Review of Known Results on Rigid Isotopy Classification
  13. Real Trigonal Curves
  14. Dessins
  15. Graph Segments
  16. ...and 18 more sections

Key Result

Proposition 1

Two generic real trigonal curves are rigidly isotopic if and only if their dessins are weakly equivalent.

Figures (23)

  • Figure 1: Chambers and walls in the space of bidegree (4,3) curves on a hyperboloid
  • Figure 2: Adjacency graphs of the chambers and walls in the spaces of plane curves of degrees 1 -- 5
  • Figure 3: Adjacency graph of chambers in the space of curves of bidegree (3,3) on a hyperboloid
  • Figure 4: Adjacency graph of chambers in the space of curves of bidegree (3,3) on an ellipsoid
  • Figure 5: Elementary moves of dessins
  • ...and 18 more figures

Theorems & Definitions (32)

  • Definition 1
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • proof
  • Definition 2
  • Definition 3
  • Proposition 3
  • Proposition 4
  • ...and 22 more