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

TL;DR

This paper studies the fixed points of a torus action on Quot schemes, revealing smoothness in low dimensions and complexity with bad singularities in higher dimensions, using incidence schemes and graph characterizations.

Contribution

It characterizes the fixed locus of the Quot scheme under torus action, showing smoothness in low dimensions and Murphy's law behavior in higher dimensions, with a decomposition into incidence schemes.

Findings

Fixed locus is smooth for d ≤ 2 or r ≤ 2.

Fixed locus exhibits Murphy's law for d ≥ 4 and r ≥ 3.

Decomposition into incidence schemes characterized by incidence graphs.

Abstract

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.