The Varchenko-Gel'fand Ring of a Cone

TL;DR

This paper generalizes the Varchenko-Gel'fand ring to cone arrangements and proves that for supersolvable arrangements, the associated graded ring is Koszul, linking algebraic properties to geometric arrangements.

Contribution

It extends the theory of the Varchenko-Gel'fand ring to cones and establishes the Koszul property for supersolvable arrangements.

Findings

Generalized Varchenko-Gel'fand ring to cone arrangements.

Proved the associated graded ring is Koszul for supersolvable arrangements.

Connected algebraic properties with geometric arrangement structures.

Abstract

For a hyperplane arrangement in a real vector space, the coefficients of its Poincar\'{e} polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise addition and multiplication. Varchenko and Gel'fand gave a simple presentation for this ring, along with a filtration and associated graded ring whose Hilbert series is the Poincar\'{e} polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes and prove a novel result for the Varchenko-Gel'fand ring of an arrangement: when the arrangement is supersolvable the associated graded ring of the arrangement is Koszul.