There are no Miyaoka--Kobayashi octics

Let $C$ be a reduced plane curve of degree $n$ with only ADE singularities. Using Kähler-Einstein metrics, Kobayashi proved an upper bound, in terms of $n$, for a sum of local invariants of the singular points of $C$. For even $n$ this bound had been obtained earlier by Hirzebruch and Ivinskis. Curves of even degree for which equality holds are called Miyaoka-Kobayashi curves, and the only known examples are three types of plane sextics. We show that there are no Miyaoka–Kobayashi curves of degree $8$, and additionally that a Miyaoka-Kobayashi curve of degree $10$ cannot be free. The key new ingredient is a new linear constraint on the singularities of such curves, obtained by differentiating Langer’s orbifold Bogomolov-Miyaoka-Yau inequality.