On the double tangent of projective closed curves

TL;DR

This paper extends a classical geometric result from planar curves to projective curves, proving positivity of bitangent counts for algebraic curves of any degree, with implications for real algebraic geometry.

Contribution

It generalizes Fabricius-Bjerre's result to projective curves and proves the positivity of signed bitangent counts for all degrees, broadening previous specific cases.

Findings

Generalization of Fabricius-Bjerre's result to $ ext{RP}^2$

Proof of positivity of signed bitangent counts for algebraic curves

Method applicable to curves of any degree

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.