Multiple-arrow incidence compactifications in dimension one at each vertex

In either word the graded flag base is a single point, since the unique flag for $(i,j)$ is $0\subset V_i\subset V$ and the unique flag for $(j,i)$ is $0\subset V_j\subset V$. There are no arrows other than $i\to j$, and $\mathrm{Hom}(V_i,V_j)=\mathbb{C}$, so $E_V={\mathbb{C}}^h$, with coordinates the $h$ arrow entries. For the word $(i,j)$ the first line is $V_i$, and stability at $r=1$ forces $x_h(V_i)\subseteq F_j^1=0$, so every arrow coordinate vanishes and the subbundle rank is zero; for the word $(j,i)$ the first line is $V_j$, stability at $r=1$ reads $x_h(F_i^1)=x_h(0)=0\subseteq V_j$ and at $r=2$ reads $x_h(V_i)\subseteq V_j$, both automatic, so every arrow coordinate is free and the rank is $h$. These are the rank counts ${\sum }_{h:i\to j}\mathrm{\#}\{(p,q):{\nu }_p=i,{\nu }_q=j,q<p\}$ of the incidence lemma Quiver full-flag incidence has a smooth projective compactification, namely $0$ for $(i,j)$ and $h$ for $(j,i)$.

The lemma's compactification adjoins a trivial line to the incidence subbundle, so the line-projective sources are ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(0\oplus \mathcal{O})$, a point over the flag point, and ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}({\mathbb{C}}^h\oplus \mathcal{O})$, and the corresponding maps to ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}({\mathbb{C}}^h\oplus \mathbb{C})={\mathbb{P}}^h$ are the point $[0:1]$ and the identity ${\mathbb{P}}^h\to {\mathbb{P}}^h$ respectively. The inverse image of the affine chart $t\mathrm{\ne }0$ is, exactly as in the lemma, the total space of the subbundle over the flag point: the point $x=0$ in the first case and all of ${\mathbb{C}}^h$ in the second. For $h=0$ both words give the two point maps, so arbitrary multiplicity including the zero case is covered.