Freeness of arrangements of lines and one conic with ordinary quasi-homogeneous singularities

The main purpose of the present paper is to provide a partial classification, performed with respect the weak-combinatorics, of free arrangements consisting of lines and one smooth conic with quasi-homogeneous ordinary singularities.

💡 Research Summary

The paper investigates the freeness of plane curve arrangements consisting of a single smooth conic together with several lines in the complex projective plane, under the restriction that all singularities are ordinary and quasi‑homogeneous of multiplicity less than five. The authors introduce the notion of “weak combinatorics” – a compact encoding of the number of components of each degree and the number of singular points of each prescribed type. For a conic‑line arrangement this reduces to a triple (d; n₂, n₃, n₄), where d is the number of lines, n₂ the number of ordinary double points (type A₁), n₃ the number of ordinary triple points (type D₄), and n₄ the number of ordinary quadruple points (type X₉).

Freeness of a reduced curve C={f=0} is traditionally defined by the saturation of its Jacobian ideal J_f, but checking this directly is cumbersome. The authors rely on two numerical criteria that are equivalent to freeness when all singularities are quasi‑homogeneous. The first, due to du Plessis and Wall, relates the minimal degree of a non‑trivial Jacobian relation, denoted r = mdr(f), to the total Tjurina number τ(C) via
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