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On the double tangent of projective closed curves

Thomas Blomme

TL;DR

The paper extends Fabricius-Bjerre’s signed count of bitangents from curves in the affine plane to curves in the projective plane by fixing a line at infinity and analyzing tangents through points of intersection with that line. It derives a projective formula for the signed count $\sigma(C)$ that incorporates the affine contributions, the number of intersections with $L_\infty$, and tangents through $C\cap L_\infty$, with key jump analyses at infinity. Applying this to real algebraic curves yields a refined invariant $\rho(\mathscr{C})=t_0(\mathscr{C})+\sigma(\mathbb{R}\mathscr{C})$ that equals $\frac{d(d-2)}{2}+\frac{a(a-2)}{2}-\sum_{T\in\mathscr{T}_\infty(\mathbb{R}\mathscr{C})}(|\mathbb{R}\mathscr{C}\cap T|-1)$, which is even and non-negative; it specializes to Larson-Vogt’s quartic count and generalizes to higher degrees, with extensions to nodal curves via Klein-type formulas. The work thus provides a unified, degree-agnostic framework for real bitangent counts and their enrichment in projective settings.

Abstract

We generalize a previous result by Fabricius-Bjerre from curves in $\mathbb R^2$ to curves in $\mathbb R P^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt and proves its positivity, conjectured by Larson-Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.

On the double tangent of projective closed curves

TL;DR

The paper extends Fabricius-Bjerre’s signed count of bitangents from curves in the affine plane to curves in the projective plane by fixing a line at infinity and analyzing tangents through points of intersection with that line. It derives a projective formula for the signed count that incorporates the affine contributions, the number of intersections with , and tangents through , with key jump analyses at infinity. Applying this to real algebraic curves yields a refined invariant that equals , which is even and non-negative; it specializes to Larson-Vogt’s quartic count and generalizes to higher degrees, with extensions to nodal curves via Klein-type formulas. The work thus provides a unified, degree-agnostic framework for real bitangent counts and their enrichment in projective settings.

Abstract

We generalize a previous result by Fabricius-Bjerre from curves in to curves in . Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt and proves its positivity, conjectured by Larson-Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.
Paper Structure (4 sections, 7 theorems, 27 equations, 5 figures)

This paper contains 4 sections, 7 theorems, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Classical Fabricius-Bjerre Theorem
  3. A projective Fabricius-Bjerre Theorem
  4. Applications to real algebraic curves

Key Result

Theorem 1

theo-projective-case Let $C$ a curve in $\mathbb{R} P^2$ transverse to $L_\infty$ with $i(C)$ flexes, $n(C)$ nodes and $a$ intersection points with $L_\infty$. Then, we have the following signed count of bitangents:

Figures (5)

  • Figure 1: Types of bitangents
  • Figure 2: Illustration of the values of the jump function at nodes, flexes and bitangents. The positive half-tangent is depicted in red, and the negative in blue. The alternative cases of passing through bitangents of types S and T are obtained by reversing the orientation of the curve and its tangents.
  • Figure 3: Description signs of tangents passing through $p_L$.
  • Figure 4: Change of value of the jump function when the curve intersects $L_\infty$: intersection points change from blue ($D_-$) to red part ($D_+$).
  • Figure 5: Change of value of the jump function when a tangent hits a point of $C\cap L_\infty$: if concavity is toward $T$ (case (a) and (b)), intersection point near $p_L$ goes from blue to red, and if concavity is toward $L_\infty$ (case (c) and (d)), intersection point near $p_L$ goes from red to blue.

Theorems & Definitions (14)

  • Theorem
  • Theorem
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • ...and 4 more