On curves of degree 10 with 12 triple points

TL;DR

This paper constructs specific degree 10 curves with numerous triple points, providing counterexamples to a conjecture and exploring degenerations to known line arrangements, advancing understanding of algebraic curve configurations.

Contribution

It introduces explicit constructions of degree 10 curves with many triple points and demonstrates their degenerations to line arrangements, challenging existing conjectures.

Findings

Constructed an irreducible rational degree 10 curve with 12 triple points.

Produced a union of three rational quartics with 19 triple points.

Established a degeneration of degree 10 curves to the dual Hesse line arrangement.

Abstract

We construct an irreducible rational curve of degree 10 in $CP^2$ 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.