New characterizations of freeness for hyperplane arrangements (Q2304214)

Let \(V\) be a vector space of dimension \(l\) over a fiels \(K\). Fix a system of coordinates \((x_1,\ldots,x_l)\) of \(V^*\). We denote by \(S=S(V^*)\) the symmetric algebra. A hyperplane arrangement \(\mathcal{A}=\{H_1,\ldots,H_n\}\) is a finite collection of hyperplanes in \(V\). For each \(H\in \mathcal{A}\) choose a linear form \(\alpha_H\) with kernel \(H\) so that \(Q(\mathcal{A})=\prod_{H\in\mathcal{A}} \alpha_H\in S\) is a defining polynomial of \(\mathcal{A}\). The Jacobian ideal of \(\mathcal{A}\) is defined as the ideal \(J(\mathcal{A})\) of \(S\) generated by \(Q(\mathcal{A})\) and all its partial derivates. The paper provides new characterizations of freeness of an arrangements using the generic initial ideal \(\mathrm{rgin}(J(\mathcal{A})\) of the Jacobian ideal. The first characterization says that a central arrangement \(\mathcal{A}=\{H_1,\ldots,H_n\}\) is free if and only iff \(\mathrm{rgin}(J(\mathcal{A})\) is \(S\) or its minimal generators include \(x_1^{n-1}\), some positive power of \(x_2\), and no monomials in \(x_3,\ldots,x_l\). Then the authors characterize freeness by looking at the sectional matrix of \(S/J(\mathcal{A})\). Futhermore they provethat for an essential and central free arrangement, \(\mathrm{rgin}(J(\mathcal{A})\) is uniquely determined by the exponents of \(\mathcal{A}\) and vice versa. Finally, the authors provide formula for when a strongly stable ideal arises as \(\mathrm{rgin}(J(\mathcal{A})\) of an essential, central and free arrangement.