The contravariant form on singular vectors of a projective arrangement

TL;DR

This paper explores the properties of the contravariant form on singular vectors in projective hyperplane arrangements, establishing a connection with affine arrangements through weights and dehomogenization.

Contribution

It introduces a new framework for understanding the contravariant form on singular vectors in projective arrangements and relates it to affine arrangements via weights and dehomogenization.

Findings

The contravariant form induces a well-defined form on singular vectors in projective arrangements.

When the sum of weights is zero, the form is isomorphic to that of an affine arrangement.

The work connects projective and affine arrangements through the contravariant form and weights.

Abstract

We define the flag space and space of singular vectors for an arrangement A of hyperplanes in projective space equipped with a system of weights a: A --> C. We show that the contravariant bilinear form of the corresponding weighted central arrangement induces a well-defined form on the space of singular vectors of the projectivization. If the sum of the weights a(H), H in A, is zero, then this form is naturally isomorphic to the restriction to the space of singular vectors of the contravariant form of any affine arrangement obtained from A by dehomogenizing with respect to one of its hyperplanes.