Moscow Mathematical Journal

Volume 26, Issue 1, January–March 2026 pp. 87–96.

On Curves of Degree 10 with 12 Triple Points

Authors: S. Yu. Orevkov (1)
Author institution:(1) IMT, Univ. de Toulouse, Toulouse, France;
Steklov Math. Inst., Moscow, Russia

Summary:

We construct an irreducible rational curve of degree 10 in $\mathbb{CP}^2$ which has 12 triple points, and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family $C_u$ of curves of degree 10 with 12 triple points which tends, as $u\to 0$, to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest.

2020 Math. Subj. Class. 14H50.

Keywords: Algebraic curve, deformation, perturbation with prescribed singulatities.

Contents Full-Text PDF