ScalingStacks

We first recall some basic definitions and facts about quiver flag varietes. We refer to [SW14] and [Prz19] for more details. Consider a quiver $Q$ with finite sets of vertices $Q_{0}$ and arrows $Q_{1}$ and source and target maps

$$s,t:Q_{1}\rightrightarrows Q_{0}.$$

Denote by $\Gamma=\mathbb{Z}_{\geq 0}Q_{0}$ and $\Gamma^{+}=\Gamma\backslash\{0\}$ the sets of (non-trivial) dimension vectors. The dimension vector of a $Q_{0}$-graded $k$-vector space $V=(V_{i})_{i\in Q_{0}}$ is $\dim(V)=(\dim(V_{i}))_{i\in Q_{0}}\in\Gamma.$ We denote by

$$\operatorname{Rep}(V)=\prod_{a\in Q_{1}}\operatorname{Hom}_{k}(V_{s(a)},V_{t(a)})$$

the vector space of quiver representations with underlying $Q_{0}$-graded vector space $V.$ We fix a dimension vector $\mathbf{d}\in\Gamma$ and let $V$ be the standard vector space with $\dim V=\mathbf{d}.$ We often abbreviate $\operatorname{Rep}(\mathbf{d})=\operatorname{Rep}(V).$ There is a a natural conjugation action on $\operatorname{Rep}(\mathbf{d})$ by the group

$$\operatorname{GL}(\mathbf{d})=\prod_{i\in Q_{0}}\operatorname{GL}(V_{i})=\prod_{i\in Q_{0}}\operatorname{GL}_{\mathbf{d}_{i}}(k).$$

A composition of a dimension vector $\mathbf{d}$ is a tuple $\underline{\mathbf{d}}=(\underline{\mathbf{d}}^{j})\in(\Gamma^{+})^{\ell_{\underline{\mathbf{d}}}}$ which sums to $\mathbf{d}.$ A composition $\underline{\mathbf{d}}$ is called complete if each $\underline{\mathbf{d}}^{j}$ is a unit vector. We write $\operatorname{Comp}(\mathbf{d})$ and $\operatorname{Compf}(\mathbf{d})$ for the set of (complete) compositions of $\mathbf{d}.$ A partial flag $\underline{V}$ of $V$ of type $\underline{\mathbf{d}}$ is a sequence of $Q_{0}$-graded $k$-vector spaces

$$0=V^{0}\subset V^{1}\subset\dots\subset V^{\ell_{\underline{\mathbf{d}}}}=V$$

such that $\dim V^{j}/V^{j-1}=\underline{\mathbf{d}}^{i}.$ We denote the smooth projective variety of such flags by $\operatorname{Fl}(V,\underline{\mathbf{d}})=\operatorname{Fl}(\underline{\mathbf{d}}).$ The partial flag $\underline{V}$ is called strictly $\rho$-stable for a quiver representation $\rho\in\operatorname{Rep}(\mathbf{d})$ if

(6.1)

$\displaystyle\rho_{a}(V^{j}_{s(a)})\subset V^{j-1}_{t(a)}\text{ for all }i=1,\dots,\ell_{\underline{\mathbf{d}}}\text{ and }a\in Q_{1}.$

We can hence consider the variety

$$\mathfrak{Q}(\underline{\mathbf{d}})=\{(\rho,\underline{V})\in\operatorname{Rep}(\mathbf{d})\times\operatorname{Fl}(\underline{\mathbf{d}})\mid\underline{V}\text{ is strictly $\rho$-stable}\}.$$

The variety $\mathfrak{Q}(\underline{\mathbf{d}})$ is smooth and has a diagonal action by $\operatorname{GL}(\mathbf{d}).$ Projection yields a $\operatorname{GL}(\mathbf{d})$-equivariant proper map

$$\mu_{\underline{\mathbf{d}}}:\mathfrak{Q}(\underline{\mathbf{d}})\to\operatorname{Rep}(\mathbf{d}).$$

For a representation $M=(V,\rho)$, the fiber $\mu_{\underline{\mathbf{d}}}^{-1}(\rho)$ is called a (partial) quiver flag variety and denoted by

$$\operatorname{Fl}(M,\underline{\mathbf{d}})=\{\underline{V}\in\operatorname{Fl}(V,\underline{\mathbf{d}})\mid\underline{V}\text{ is strictly $\rho$-stable}\}.$$