  Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where
$V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation
stability} and {\em multiplicity stability} are two related notions of when the
sequence $V_n$ has a limit. An important source of stability phenomena arises
in the case where $V_n$ is the $d^{th}$ homology group (for fixed $d$) of the
configuration space of $n$ distinct points in some fixed topological space $X$.
We replace these configuration spaces with the variety $X_{n,k}$ of {\em
spanning configurations} of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in
$\mathbb{C}^k$ which satisfy $\ell_1 + \cdots + \ell_n = \mathbb{C}^k$ as
vector spaces. We study stability phenomena for the homology groups
$H_d(X_{n,k})$ as the parameter $(n,k)$ grows.
