On the Jacobian ideal of central arrangements

TL;DR

This paper investigates the relationship between the Jacobian ideal and the ideal generated by hyperplane defining forms in central arrangements, exploring properties and conjectures, with extensions to higher degree forms under transversality conditions.

Contribution

It establishes new insights into the structure of Jacobian and hyperplane form ideals in central arrangements and proposes related conjectures, extending results to higher degree forms under specific conditions.

Findings

Analyzed the properties of Jacobian and hyperplane form ideals.

Proved results relating the two ideals in central arrangements.

Extended some results to higher degree forms with transversality.

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 $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial $f:=l_1\cdots l_m\in R$. 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 $l_1,\ldots, l_m$. 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.