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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 thestandard action on $\mathbb{A}^d$. We analyze the fixed locus of the Quotscheme under this action. In particular we show that for $d \leq 2$ or $r \leq2$, this locus is smooth, and that for $d \geq 4$ and $r \geq 3$ it satisfiesMurphy's law as introduced by Vakil, meaning that it has arbitrarily badsingularities. These results are obtained by giving a decomposition of thefixed locus into connected components, and identifying the components withincidence schemes of subspaces of $\mathbb{P}^{r-1}$. We then obtain acharacterization of the incidence schemes which occur, in terms of their graphsof incidence relations.

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