there are only finitely many $P$-invariant linear subspaces $Z$ on which to apply Kempf’s construction.

Quiver loci
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A [[****]{}quiver]{} $Q=(Q_0,Q_1)$ is a finite directed graph, which consists of a set $Q_0$ of vertices and a set $Q_1$ of directed edges or arrows, such that each arrow $a\in Q_1$ has a [[****]{}tail]{} $ta\in Q_0$ and a [[****]{}head]{} $ha\in Q_0$. A [[****]{}representation]{} $V$ of $Q$ is a choice of a vector space $V(i)$ for each vertex $i\in Q_0$ and a linear map $V(a)\in{\mathrm{Hom}}(V(ta),V(ha))$ for each arrow $a\in Q_1$. There are obvious notions of isomorphism, direct sum, and indecomposable, for representations of $Q$. The [[****]{}dimension vector]{} of $V$ is the map $Q_0\rightarrow {{\mathbb N}}$ defined by $i\mapsto \dim V(i)$. Fix a dimension vector $d:Q_0\rightarrow{{\mathbb N}}$ and define $$GL := GL(Q,d) = \prod_{i\in Q_0} GL({\mathbb{C}}^{d(i)}), \qquad
{\mathrm{Hom}}:= {\mathrm{Hom}}(Q,d) = \prod_{a\in Q_1} {\mathrm{Hom}}({\mathbb{C}}^{d(ta)},{\mathbb{C}}^{d(ha)}).$$ A typical element of ${\mathrm{Hom}}$ is denoted $V$, and for $a\in Q_1$ the $a$ component is denoted $V(a)$. The notation comes from thinking of $V$ as a functor from the free category on $Q$ to the category [**Vec**]{}. The group $GL$ acts linearly on ${\mathrm{Hom}}$ by change of basis: $(g\cdot V)(a) = g(ta) V(a) g(ha)^{-1}$ for all $g\in GL$, $V\in
{\mathrm{Hom}}$, 