Bitangents of real algebraic curves: signed count and constructions

2024/02/06 by Thomas Blomme, Blomme, Thomas, Erwan Brugallé +3 · 1 citation
Computer Science · Engineering · Mathematics · #14H50 #14N05 #14N15 #14P25 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2402.03993

openalex publication_date 2024/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic real algebraic curve of even degree d has at least (d(d-2))/(2) real bitangents. Next we explain how to locate (real) bitangents of a (real) perturbation of a multiple (real) conic in ℂP². As main applications, we exhibit a real sextic with a total of 318 real bitangents and 6 complex ones, and perform asymptotical constructions that give the best, to our knowledge, number of real bitangents of real algebraic plane curves of a given degree.

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• The Evolution of Enumerative Geometry: A Narrative from Classical Problems to Enriched Invariants

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