One-color quiver full flags need not be semismall

There are no arrows, so $E_V=0$, and the stability conditions of the incidence lemma Quiver full-flag incidence has a smooth projective compactification are vacuous; the lemma gives $X_{\nu }=\mathrm{Flag}(V)$ as the total space of the rank-zero subbundle, of dimension $n(n-1)\mathrm{/}2$, and the map ${\pi }_{\nu }$ is the constant map to the point $E_V=0$, which is proper since it is projective.

The semismall inequality on a target stratum $T$ with fibre dimension $d$ is $2d+\dim T\le \dim X_{\nu }$; here there is just the point stratum. The image of ${\pi }_{\nu }$ is the single point, and its single fiber is all of $X_{\nu }$, so the defect $2\dim (\text{fiber})+\dim (\text{image})-\dim X_{\nu }$ equals $2\cdot n(n-1)\mathrm{/}2+0-n(n-1)\mathrm{/}2=n(n-1)\mathrm{/}2$, which is positive for $n\ge 2$; hence this proper incidence map is not semismall. For $n=0$ or $n=1$ the flag space is a point and the defect is zero, so the non-semismall boundary is exactly $n\ge 2$, and a general quiver-flag decomposition proof must allow this case.