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Algebraic curves, rich points, and doubly-ruled surfaces

Larry Guth, Joshua Zahl

TL;DR

Let $k$ be a field and $\\mathcal{L}$ a collection of $n$ irreducible space curves in $k^3$ of degree at most $D$ with $n$ small relative to the characteristic. The paper proves a dichotomy: either at most $O(n^{3/2})$ points lie on two or more curves from $\\mathcal{L}$, or there exists an irreducible surface $Z$ of degree $\le 100D^2$ containing at least $A$ curves from $\\mathcal{L}$ such that the hat-$Z$ is doubly ruled by the curves from a prescribed constructible family. A central tool is a generalized flecnodal framework built from $r$-jets and constructible conditions, enabling a polynomial–style detection of doubly ruled structures and a mechanism, called 'sufficiently tangent implies trapped', that forces curves into the ruling surface. The authors develop an affine Chow-Variety approach to parameterize curves, prove degree bounds for doubly ruled surfaces, and extend Guth–Katz-type incidence results from lines to bounded-degree curves across fields with appropriate characteristic constraints. Overall, the work provides new algebraic techniques for incidence geometry with potential applications to broader counting problems and structural theorems in algebraic combinatorics.

Abstract

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\!\!<(\operatorname{char}(k))^2$ or $\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $Ω(n^{1/2})$ curves from $\mathcal{L}$ must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.

Algebraic curves, rich points, and doubly-ruled surfaces

TL;DR

Let be a field and a collection of irreducible space curves in of degree at most with small relative to the characteristic. The paper proves a dichotomy: either at most points lie on two or more curves from , or there exists an irreducible surface of degree containing at least curves from such that the hat- is doubly ruled by the curves from a prescribed constructible family. A central tool is a generalized flecnodal framework built from -jets and constructible conditions, enabling a polynomial–style detection of doubly ruled structures and a mechanism, called 'sufficiently tangent implies trapped', that forces curves into the ruling surface. The authors develop an affine Chow-Variety approach to parameterize curves, prove degree bounds for doubly ruled surfaces, and extend Guth–Katz-type incidence results from lines to bounded-degree curves across fields with appropriate characteristic constraints. Overall, the work provides new algebraic techniques for incidence geometry with potential applications to broader counting problems and structural theorems in algebraic combinatorics.

Abstract

We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let be a field and let be a collection of space curves in , with or . Then either A) there are at most points in hit by at least two curves, or B) at least curves from must lie on a bounded-degree surface, and many of the curves must form two "rulings" of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.

Paper Structure

This paper contains 35 sections, 60 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Proof sketch and main ideas
  3. Previous work
  4. Thanks
  5. Constructible sets
  6. The Chow variety of algebraic curves
  7. Surfaces doubly ruled by curves
  8. Statement of the theorem
  9. Curves and complete intersections
  10. Local Rings, Intersection multiplicity and Bézout
  11. Local rings
  12. The Bézout theorem
  13. Curve-surface tangency
  14. Differentiating along a curve
  15. Defining curve-surface tangency
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\mathcal{L}$ be a collection of $n$ lines in $\mathbb{R}^3$. Let $A\geq 100n^{1/2}$ and suppose that there are $\geq 100An$ points $p\in\mathbb{R}^3$ that are incident to at least two points of $\mathcal{L}$. Then there exists a plane or regulus $Z\subset\mathbb{R}^3$ that contains at least $A$

Theorems & Definitions (122)

  • Theorem 1.1: Guth-Katz GuthKatz
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4: Chevalley; see harris, Theorem 3.16 or matsumura, Chapter 2.6, Theorem 6
  • Definition 2.5
  • Proposition 3.1: Properties of the (projective) Chow variety
  • Lemma 3.2: Properties of the (affine) Chow variety of degree $D$ curves
  • Lemma 3.3: Properties of the (affine) Chow variety
  • ...and 112 more