Murphy's law on a fixed locus of the Quot scheme

Let $T:={\mathbb{G}}_m^d$ be the torus acting on the Quot scheme of points ${\coprod }_n{\mathrm{Q}\mathrm{u}\mathrm{o}\mathrm{t}}_{{\mathcal{O}}^r\mathrm{/}{\mathbb{A}}^d\mathrm{/}\mathbb{Z}}^n$ 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 $d\le 2$ or $r\le 2$, this locus is smooth, and that for $d\ge 4$ and $r\ge 3$ 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.