On the Numerical Terao Conjecture

Abstract

We prove that the Numerical Terao Conjecture holds for even-degree conic-line arrangements having only ADE singularities. We then show that the conjecture fails in the broader quasi-homogeneous setting once ordinary quadruple points are admitted, by constructing a degree-nine counterexample, each consisting of seven lines and one smooth conic, with the same weak combinatorics $$W(\mathcal{CL}) = (7,1;\,8A_1+D_4+4X_9).$$ We show that one curve is free with exponents $(4,4)$, whereas the other is nearly free with exponents $(3,6)$. This yields a counterexample to the strongest known formulation of the Numerical Terao Conjecture.