On the Jacobian ideal of an almost generic hyperplane arrangement

Abstract:

Let $\mathcal {A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb {K}^n$ over a field $\mathbb {K}$ of characteristic zero and let $l_1,\ldots , l_m\in R≔\mathbb {K}[x_1,\ldots ,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f≔l_1\cdots l_m\in R$. The focus of the paper is on the ideal $J_f\subset R$ generated by the partial derivatives of $f$. We conjecture that $J_f$ is a minimal reduction of the ideal $\mathbb {I}\subset R$ generated by the $(m-1)$-fold products of distinct forms among $l_1,\ldots , l_m$. We prove this conjecture for an almost generic $\mathcal {A}$ (i.e., any $n-1$ among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for $n=3$. We also conjecture that $J_f$ is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for $n=3$. In the sequel we explain the tight relationship between the two ideals $J_f, \mathbb {I}\subset R$; in particular, we show that in the generic case $(J_f)^{\text {sat}}=\mathbb I$. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of $R/J_f$.