Symmetry of $f$-vectors of toric arrangements in general position and some applications

Abstract

A toric hyperplane is the preimage of a point $x \in S^1$ of a continuous surjective group homomorphism $\theta: \mathbb{T}^n \to S^1$. A finite hyperplane arrangement is a finite collection of such hyperplanes. In this paper, we study the combinatorial properties of finite hyperplane arrangements on $\mathbb{T}^n$ which are spanning and in general position. Specifically, we describe the symmetry of $f$-vectors arising in such arrangements and a few applications of the result to count configurations of hyperplanes.