PreprintCC BY 4.0arXiv
There are no Miyaoka--Kobayashi octics
Abstract
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.
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BibTeX
@misc{dimca2026there,
title = {{There are no Miyaoka--Kobayashi octics}},
author = {Alexandru Dimca and Piotr Pokora},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28135},
eprint = {2610.28135},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28135},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - There are no Miyaoka--Kobayashi octics
AU - Alexandru Dimca
AU - Piotr Pokora
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28135
UR - https://arxiv.org/abs/2610.28135
AB - 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.
ER -