calls a [[****]{}collapsing]{} of $G\times^P Z$. This space is the associated fiber bundle with fiber $Z$ over the homogeneous projective variety $G/P$. The map $\kappa$ is proper (it factors as $G\times^P Z {\operatorname*{\hookrightarrow}}G/P \times Y {\mathop{\twoheadrightarrow}}Y$), hence its image $G\cdot Z$ is closed. When $\kappa$ is birational, $\kappa$ serves as a resolution of singularities of the variety $G\cdot Z$.

Since $G\cdot Z$ is a $G$-invariant subvariety of ${\mathrm{Hom}}$, it has a [*multidegree*]{} $[G\cdot Z]$, which can be defined as the associated $G$-equivariant cohomology class (or Chow class) in the ring $H^*_G(Y)$. Our first result, Theorem \[thm:multidegree\] below, is a formula for $[G\cdot Z]$. In Section \[sec:mdegs\] we recall the properties we need of multidegrees. If $T\leq P$ is a maximal torus of $G$, then $H^*_G(Y)$ naturally includes into $H^*_T(Y) = {\mathrm{Sym}}^\bullet(T^*)$, so it is enough to compute $[G\cdot Z]$ as a $T$-equivariant cohomology class. Since $Z$ is $P$-invariant and hence $T$-invariant, it too has a $T$-multidegree $[Z] \in {\mathrm{Sym}}^\bullet(T^*)$. This $[Z]$ turns out to be particularly simple for $Z$ a linear subspace: $[Z]$ is the formal product of the $T$-weights in $Y/Z$, each of which lives in the weight lattice $T^*$ of $T$. If none of the $T$-weights on $Y$ 