  Let $T := \mathbb{G}_m^d$ be the torus acting on the Quot scheme of points
$\coprod_n \mathrm{Quot}_{\mathcal{O}^r/\mathbb{A}^d/\mathbb{Z}}^n$ via the
standard action on $\mathbb{A}^d$. We analyze the fixed locus of the Quot
scheme under this action. In particular we show that for $d \leq 2$ or $r \leq
2$, this locus is smooth, and that for $d \geq 4$ and $r \geq 3$ it satisfies
Murphy's law as introduced by Vakil, meaning that it has arbitrarily bad
singularities. These results are obtained by giving a decomposition of the
fixed locus into connected components, and identifying the components with
incidence schemes of subspaces of $\mathbb{P}^{r-1}$. We then obtain a
characterization of the incidence schemes which occur, in terms of their graphs
of incidence relations.
