The one-loop quiver extension correspondence in dimension two

A one-loop representation of dimension one is a scalar, and change of basis by ${\mathrm{GL}}_1$ fixes it because conjugation by a scalar is trivial. A dimension-two representation is a $2\times 2$ matrix with conjugation action. Preserving a chosen line makes the matrix upper triangular in an adapted basis, as displayed.

For this quiver the off-diagonal rank is $e=1$ and the unipotent vertex-block rank is $u=1$; hence $q$ has relative stack dimension $e-u=0$, agreeing with $-{\chi }_Q(1,1)=0$. For the splitting change represented by $t\in \mathrm{Hom}(\mathbb{C},\mathbb{C})$, the change-of-basis matrix $g_t=\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)$ acts by $A\mapsto g_{t}A{g}_t^{-1}$, sending $c$ to $c+(\mu -\lambda )t$. Thus when $\lambda \mathrm{\ne }\mu$ the extension coordinate is removed by a unique splitting change, whereas when $\lambda =\mu$ it is unchanged; this is consistent with a zero-dimensional stack morphism, not a claim that all extension fibers are points.

The fiber of $p$ at a scalar matrix $\lambda I$ is all of ${\mathbb{P}}^1$, so it is positive-dimensional. For a diagonal matrix with distinct eigenvalues, an invariant line must be an eigenspace, giving exactly two points. For a non-scalar Jordan matrix $\lambda I+N$ with $N^2=0$ and $\mathrm{rank}N=1$, an invariant line is $N$-stable; the nilpotent map induced on a one-dimensional line is zero, so the line lies in $\ker N$, which is the unique invariant line. These fibers agree with the closed incidence projection to ${\mathrm{Mat}}_2(\mathbb{C})\times {\mathbb{P}}^1$ and do not affect properness.

The incidence equations $A(L)\subseteq L$ are closed in ${\mathrm{Mat}}_2(\mathbb{C})\times {\mathbb{P}}^1$, so the forgetful map $p$ is projective and proper by [F2]. The example includes a loop and checks why stack-relative dimension and properness must be read from the specified correspondence, not from finiteness of its fibers. AC is inherited through [A1].