On Poincaré polynomials for plane curves with quasi-homogeneous singularities

Abstract

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincaré 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.

10 pages, this is the final version, incorporating the referee's comments, to be published in the Bulletin of the London Mathematical Society

References in corpus (3)

• On $3$-syzygy and unexpected plane curves
• Freeness of arrangements of lines and one conic with ordinary quasi-homogeneous singularities
• On free and nearly free arrangements of conics admitting certain $\mathrm{ADE}$ singularities

Cited by in corpus (2)

• On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities
• On plus-one generated arrangements of plane conics