# Spanning subspace configurations

## TL;DR

This paper computes the integral cohomology of the moduli space of spanning configurations in complex vector spaces, generalizing classical flag variety results and connecting to symmetric function theory through spanning line configurations.

## Contribution

It provides a new cohomological description of spanning configuration moduli spaces, extending classical and recent geometric frameworks.

## Key findings

- Derived the integral cohomology of spanning configuration moduli spaces.
- Unified classical flag variety cohomology with spanning line configuration results.
- Connected geometric structures to the Haglund-Remmel-Wilson Delta Conjecture.

## Abstract

A {\em spanning configuration} in the complex vector space $\mathbb{C}^k$ is a sequence $(W_1, \dots, W_r)$ of linear subspaces of $\mathbb{C}^k$ such that $W_1 + \cdots + W_r = \mathbb{C}^k$. We present the integral cohomology of the moduli space of spanning configurations in $\mathbb{C}^k$ corresponding to a given sequence of subspace dimensions. This simultaneously generalizes the classical presentation of the cohomology of partial flag varieties and the more recent presentation of a variety of spanning line configurations defined by the author and Pawlowski. This latter variety of spanning line configurations plays the role of the flag variety for the Haglund-Remmel-Wilson Delta Conjecture of symmetric function theory.