A basis for the cohomology of compact models of toric arrangements

TL;DR

This paper constructs explicit monomial bases for the integer cohomology rings of compact models of toric arrangements, linking combinatorial structures with geometric and algebraic properties.

Contribution

It introduces a combinatorial framework for describing cohomology bases of toric arrangement models, including computational tools and special cases related to root systems of type A.

Findings

Explicit monomial bases for cohomology rings are provided.

Connections between combinatorial objects and geometric structures are established.

Examples demonstrate the computational approach using SageMath.

Abstract

In this paper we find monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into play: building sets, nested sets, and the fan of a suitable toric variety. We provide some examples computed via a SageMath program and then we focus on the case of the toric arrangements associated with root systems of type A. Here the combinatorial description of our basis offers a geometrical point of view on the relation between some Eulerian statistics on the symmetric group.