$\mathcal{Q}$-conic arrangements in the complex projective plane

TL;DR

This paper investigates the geometric properties of arrangements of smooth conics with specific singularities in the complex projective plane, revealing they are never free and establishing combinatorial constraints.

Contribution

It introduces the concept of $ ext{Q}$-conic arrangements, proves their non-freeness, and derives combinatorial restrictions on their configurations.

Findings

$ ext{Q}$-conic arrangements are never free.

Provides combinatorial constraints for these arrangements.

Advances understanding of conic arrangements in algebraic geometry.

Abstract

We study the geometry of $\mathcal{Q}$-conic arrangements in the complex projective plane. These are arrangements consisting of smooth conics and they admit certain quasi-homogeneous singularities. We show that such $\mathcal{Q}$-conic arrangements are never free. Moreover, we provide combinatorial constraints of the weak combinatorics of such arrangements.