Spanning subspace configurations

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.