Equivariant cohomology of Grassmannian spanning lines

Given integers $n\ge k\ge d$, let $X_{n,k,d}$ be the moduli space of $n$-tuples of lines $({\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_n)$ in ${\mathbb{C}}^k$ such that ${\mathrm{\ell }}_1+\cdots +{\mathrm{\ell }}_n$ 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.