Equivariant cohomology of Grassmannian spanning lines

TL;DR

This paper computes the torus-equivariant cohomology of a moduli space of line configurations in complex space, using combinatorial deformation theory to provide a quotient presentation.

Contribution

It introduces a new quotient presentation of the equivariant cohomology of the space of line tuples with fixed span dimension, connecting geometric and combinatorial methods.

Findings

Provides explicit quotient presentation of equivariant cohomology

Connects orbit harmonics method with geometric cohomology calculations

Enhances understanding of moduli spaces of line configurations

Abstract

Given integers $n \geq k \geq d$, let $X_{n,k,d}$ be the moduli space of $n$-tuples of lines $(\ell_1, \dots, \ell_n)$ in $\mathbb{C}^k$ such that $\ell_1 + \cdots + \ell_n$ has dimension $d$. We give a quotient presentation of the torus-equivariant cohomology of $X_{n,k,d}$. The form of this presentation, and in particular the torus parameters appearing therein, will arise from the orbit harmonics method of combinatorial deformation theory.