ScalingStacks

Since $M=(V,\rho)$ is nilpotent, we have $\operatorname{soc}(M)=\ker(\rho).$ Hence, for every flag $\underline{V}=(0\subset V^{1}\subset\dots)$ that is strictly $\rho$-stable we have $V_{1}\subset\operatorname{soc}(M),$ see (6.1). We hence get a natural map

(6.2)

$\displaystyle p:\operatorname{Fl}(M,\underline{\mathbf{d}})\to\operatorname{Gr}(\operatorname{soc}(M),\underline{\mathbf{d}}^{1}),\,\underline{V}\mapsto V_{1}$

from the partial quiver flag variety to the Grassmannian of subspaces of $\operatorname{soc}(M)$ with dimension vector $\underline{\mathbf{d}}^{1}=\dim V^{1}.$ We construct an affine paving of $\operatorname{Fl}(M,\underline{\mathbf{d}})$ using $p$ inductively. We will show that the preimage under $p$ of a $B$-orbit in the Grassmannian has an affine paving and then deduce the claim.