Terao's freeness conjecture for arrangements of 15 and 16 lines

Abstract

We prove that Terao's freeness conjecture holds for arrangements of 15 and of 16 lines in the complex projective plane. Together with the theorem of Barakat and Kühne for at most 14 hyperplanes, this shows that freeness of a central arrangement of at most 16 planes in $\mathbb{C}^3$ is determined by its intersection lattice. The proof has a theoretical and a computational part. Combining published combinatorial criteria for freeness (due to Abe, Dimca, Dimca-Kühne-Pokora and Yoshinaga) with Bojanowski's Hirzebruch-type inequality and an elementary counting identity, we show that a counterexample would have exponents $(6,8)$ or $(7,7)$ for 15 lines and $(7,8)$ for 16 lines, and an intersection lattice subject to strong numerical constraints. We then enumerate all 22,544,280 simple rank 3 matroids satisfying these constraints, up to isomorphism, and determine the complex realization space of each of them exactly over $\mathbb{Q}$. Exactly eleven of them are realizable over $\mathbb{C}$, and on each of these the freeness of the realizations is constant. As a by-product we classify the free arrangements of 15 and 16 lines that are not divisionally free, have maximal multiplicity smaller than the smaller nontrivial exponent, and have at least four intersection points on every line.