The generic splitting type of the logarithmic bundle of an arrangement of at most sixteen lines is combinatorial

Source abstract

Let $\mathcal{A}$ be an arrangement of $d$ lines in $\mathbb{P}^2_\mathbb{C}$ and let $\mathcal{T}\langle\mathcal{A}\rangle$ be its bundle of logarithmic vector fields. On a general line, $\mathcal{T}\langle\mathcal{A}\rangle$ splits as $\mathcal{O}(-k)\oplus\mathcal{O}(k+1-d)$ with $k=\min\{\mathrm{mdr}(\mathcal{A}),\lfloor\frac{d-1}{2}\rfloor\}$, where $\mathrm{mdr}$ is the minimal degree of a Jacobian syzygy. We prove that $k$ is determined by the intersection lattice whenever $d\le16$, answering in this range a question of Cook, Harbourne, Migliore and Nagel. Since freeness can be read off from $k$ and $c_2(\mathcal{T}\langle\mathcal{A}\rangle)$, this strengthens Terao's freeness conjecture in this range; in particular we obtain a proof of Terao's conjecture for $14$ lines that uses it only for $13$ lines. For any number $d$ of lines, it follows that $\mathrm{mdr}$ is determined by the lattice whenever $\mathrm{mdr}\le6$ and $2\mathrm{mdr}\le d-3$, and that Terao's conjecture holds for every free arrangement whose smaller exponent is at most $7$. The proof reduces the question to critical arrangements: $2s+3$ lines, each containing at most $s+1$ intersection points, with $\mathrm{mdr}=s$. For these, the inflection polynomial of the minimal derivation factors as $Q_\mathcal{A} R$ with $R\in H^0(\mathcal{I}_Z^2(s-3))$, where $Z$ is the zero scheme of the derivation, of length $ν=c_2(\mathcal{T}\langle\mathcal{A}\rangle)-s(s+2)$. For $s\le5$ this forces $ν\le1$, and Chern class arguments then reduce everything to Terao's conjecture for $13$ lines. For $s=6$ we show $ν\le3$, using a Riccati equation along multiple components of $R$, Ziegler's multirestriction, and local computations verified by computer algebra; an exhaustive enumeration of the 26,265,318 candidate lattices with $ν\in\{2,3\}$ that satisfy Bojanowski's inequality completes the proof.