On the Jacobian ideal of central arrangements

Abstract

Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space ${\mathbb{K}}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\dots ,l_m\in R:=\mathbb{K}[x_1,\dots ,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f:=l_1\cdots l_m\in R$. Let $J_f$ denote the ideal generated by the partial derivatives of $f$ and let $\mathbb{I}$ designate the ideal generated by the $(m-1)$-fold products of $l_1,\dots ,l_m$. This paper is centered on the relationship between the two ideals $J_f,\mathbb{I}\subset R$, their properties and two conjectures related to them. Some parallel results are obtained in the case of forms of higher degrees provided they fulfill a certain transversality requirement.

20 pages