On Poincar\'e polynomials for plane curves with quasi-homogeneous singularities

TL;DR

This paper introduces a combinatorial Poincaré polynomial for conic-line arrangements with ordinary singularities, proves a factorization property for free arrangements with quasi-homogeneous singularities, and provides constraints for such arrangements.

Contribution

It defines a new combinatorial invariant and establishes a factorization theorem for free arrangements with quasi-homogeneous singularities, extending known results in the field.

Findings

Proposes a combinatorial Poincaré polynomial for conic-line arrangements.

Proves a Terao-type factorization for free arrangements with quasi-homogeneous singularities.

Provides combinatorial constraints for free d-arrangements with specific singularities.

Abstract

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of such a polynomial over the rationals under the assumption that our conic-line arrangements are free and admit ordinary quasi-homogeneous singularities. Then we focus on the so-called $d$-arrangements in the plane. In particular, we provide a combinatorial constraint for free $d$-arrangements admitting ordinary quasi-homogeneous singularities.