Semistable loci of finite-quiver representations are open

Work over $\bar{k}$. By [F2] and [F7], the Plucker equations define a closed projective subscheme whose $\bar{k}$-points are the $r$-planes. On a chart $p_I\mathrm{\ne }0$, normalize $p_I=1$ and use the signed one-outside coordinates as the entries of a matrix $A$. The Plucker relation with tuples $J\setminus \{j\}$ and $I\cup \{j\}$ expresses each $p_J$, for $j\in J\setminus I$, using coordinates with fewer indices outside $I$ and one-outside coordinates. Induction identifies every $p_J$ in the quotient chart ring with the corresponding maximal minor of $(I_r\mid A)$; conversely those minors satisfy every Plucker relation. Hence the chart ring is $\bar{k}[A]$. The row spans of $(I_r\mid A)$ give a rank-$r$ subbundle of the trivial bundle, and its quotient is locally free; the pivot changes on overlaps are invertible and glue these bundles and charts. Thus the equations construct the Grassmannian scheme with its closed Plucker embedding. The endpoint cases $r=0,n$ are points. For the Segre equations in [F3], on $z_{i_0{j}_0}\mathrm{\ne }0$ normalize $z_{i_0{j}_0}=1$ and put $u_i=z_{i{j}_0}$ and $v_j=z_{i_{0}j}$, with $u_{i_0}=v_{j_0}=1$. The minor relations give $z_{ij}=u_i{v}_j$ in the quotient ring, and substitution and coordinate extraction are inverse maps with $\bar{k}[u_i(i\mathrm{\ne }{i}_0),v_j(j\mathrm{\ne }{j}_0)]$. These charts construct the closed Segre embedding. Iterating it embeds the finite product $P_{\mathbf{e}}={\prod }_i\mathrm{Gr}(e_i,k^{d_i}{)}_{\bar{k}}$ as a closed subscheme of one projective space; if $Q_0$ is empty, $P_{\mathbf{e}}$ is a point. The chart ring surjections remain surjective after adjoining the arrow-matrix coordinates, so ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$ is closed in a relative projective space over ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$. The chart calculations show these projective subschemes are reduced with the point loci in [F2] and [F3]; [A2] records their classical projective coordinate interpretation over $\bar{k}$.

For each subvector $\mathbf{e}$ with $0<\mathbf{e}\le \mathbf{d}$ and ${\varphi }_Z(\mathbf{e})>{\varphi }_Z(\mathbf{d})$, impose in ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$, for each arrow $a:i\to j$, that the composition ${\mathcal{S}}_i⟶k^{d_i}\stackrel{A_a}{\to }{k}^{d_j}⟶{\mathcal{Q}}_j$ vanish. Here ${\mathcal{S}}_i$ and ${\mathcal{Q}}_j$ are the tautological subbundle and quotient bundle constructed in step 1.1; their fibers at $U_i,U_j$ are $U_i$ and $k^{d_j}\mathrm{/}{U}_j$. On the charts from step 1.1 these are polynomial equations, so their common zero locus $I_{\mathbf{e}}$ is closed. Its geometric points are exactly pairs consisting of an arrow-matrix tuple and a subrepresentation of dimension $\mathbf{e}$. The closed inclusion $I_{\mathbf{e}}\hookrightarrow {\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}\times P_{\mathbf{e}}$ composes with the relative Plucker–Segre closed immersion of step 1.1, so the projection to ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$ is projective by [F4].

By [A1] and [F5], the projection in step 2.1 is proper; [F6] then makes its image closed in ${\mathrm{Rep}}_{\mathbf{d}}(Q{)}_{\bar{k}}$. Its $\bar{k}$-points are exactly the representations with a destabilizing subrepresentation of dimension $\mathbf{e}$.

There are finitely many subvectors $\mathbf{e}$ with $0<\mathbf{e}\le \mathbf{d}$. The unstable locus is the finite union of the closed images from step 3.1 over the subvectors satisfying the strict phase inequality. Its complement is therefore open and, by [F1], is precisely ${\mathrm{Rep}}_{\mathbf{d}}^{ss}(Q,Z)$. If $\mathbf{d}=0$ the union is empty and the whole one-point representation space is open.

The two declared axiom branches are [A2], used in step 1.1 through [F7] for the scheme-theoretic projective constructions, and [A1], used in step 3.1 through [F5] for properness. The semistability and HN counting arguments do not otherwise use AC.