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Given positive integers $d \leq k \leq n$, we consider the moduli space X_{n,k,d} of lines $(\ell_1,\dots,\ell_n)$ in $\mathbb{C}^k$ such that $\ell_1 + \dots + \ell_n$ has vector space dimension $d$. This is a generalization of the "spanning line configurations" of Pawlowski-Rhoades. The space $X_{n,k,d}$ carries an action of the torus $(\mathbb{C}^*)^k$, and we present the equivariant cohomology of $X_{n,k,d}$ using the orbit harmonics technique from combinatorial deformation theory. This solves a problem of Pawlowski-Rhoades, and suggests a connection between orbit harmonics and equivariant cohomology.