On the double tangent of projective closed curves

Abstract

We generalize a previous result by Fabricius–Bjerre [Math. Scand. 11 (1962), no. 2, 113–116] from curves in $R^2$ to curves in $R{P}^2$. Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson–Vogt [Res. Math. Sci. 8 (2021), no. 26] and proves its positivity, conjectured by Larson–Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.