The Jacobian ideal of a hyperplane arrangement

TL;DR

This paper proves that a hyperplane arrangement can be uniquely reconstructed from its Jacobian ideal, which is generated by the partial derivatives of its defining polynomial, providing a new algebraic characterization.

Contribution

It establishes a novel result linking the Jacobian ideal to the geometric structure of hyperplane arrangements, enabling reconstruction from algebraic data.

Findings

Arrangement can be reconstructed from its Jacobian ideal

Jacobian ideal uniquely determines the arrangement

Provides algebraic tools for studying arrangements

Abstract

The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.