Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounding the signed count of real bitangents to plane quartics

Mario Kummer, Stephen McKean

TL;DR

Problem: bound the signed count $s_L(Q)$ of real bitangents to a real smooth plane quartic $Q$ relative to an auxiliary line $L$. Approach: combine $\mathbb{A}^1$-enumerative geometry with real plane geometry and convexity to analyze the $\operatorname{QType}_L(B)$ and the grates $g_L(B)$, then study split bitangents across connected components of $Q(\mathbb{R})$. Contributions: prove $s_L(Q) \le 8$, hence $s_L(Q) \in \{0,2,4,6,8\}$, aligning with existing lower bounds and examples. Significance: connects enriched enumerative methods to real algebraic geometry and convex-geometry arguments, delivering a robust universal upper bound and clarifying the structure of real bitangent configurations.

Abstract

Using methods from enriched enumerative geometry, Larson and Vogt gave a signed count of the number of real bitangents to real smooth plane quartics. This signed count depends on a choice of a distinguished line. Larson and Vogt proved that this signed count is bounded below by 0, and they conjectured that the signed count is bounded above by 8. We prove this conjecture using real algebraic geometry, plane geometry, and some properties of convex sets.

Bounding the signed count of real bitangents to plane quartics

TL;DR

Problem: bound the signed count of real bitangents to a real smooth plane quartic relative to an auxiliary line . Approach: combine -enumerative geometry with real plane geometry and convexity to analyze the and the grates , then study split bitangents across connected components of . Contributions: prove , hence , aligning with existing lower bounds and examples. Significance: connects enriched enumerative methods to real algebraic geometry and convex-geometry arguments, delivering a robust universal upper bound and clarifying the structure of real bitangent configurations.

Abstract

Using methods from enriched enumerative geometry, Larson and Vogt gave a signed count of the number of real bitangents to real smooth plane quartics. This signed count depends on a choice of a distinguished line. Larson and Vogt proved that this signed count is bounded below by 0, and they conjectured that the signed count is bounded above by 8. We prove this conjecture using real algebraic geometry, plane geometry, and some properties of convex sets.
Paper Structure (3 sections, 18 theorems, 9 equations, 4 figures)

This paper contains 3 sections, 18 theorems, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A few results on plane geometry and convex sets
  3. Proving the conjecture

Key Result

Theorem 1.1

Let $Q\subset\mathbb{P}^2$ be a real smooth plane quartic. For any admissible line $L$, the signed count $s_L(Q)$ is at most 8. In particular, $s_L(Q)\in\{0,2,4,6,8\}$.

Figures (4)

  • Figure 1: Support of lines through opposite edges of a quadrilateral
  • Figure 2: Grates meeting two components
  • Figure 3: Quadrilateral from positive grates
  • Figure 4: Meeting two negative grates

Theorems & Definitions (37)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 27 more