Cohomology of the Quot scheme of an infinite affine space

TL;DR

This paper investigates the cohomology of Quot schemes of points on affine spaces, identifying explicit loci and their properties, and confirms a conjecture for the case d=2, advancing understanding of these geometric objects.

Contribution

It computes the cohomology of specific loci in Quot schemes and analyzes their structure, including the ind-scheme limit, providing new classification results and confirming a conjecture.

Findings

Cohomology of explicit loci in Quot schemes computed

Complement of loci has diverging codimension as n increases

Confirmed Pandharipande's conjecture for d=2 case

Abstract

We study the Quot scheme of points $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$. We exhibit and compute the cohomology of explicit loci in $\mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r})$, whose complement has codimension diverging to infinity as $n\rightarrow \infty$. In the case $1<r<\frac{d+1}{2}$ this loci is an irreducible component. The main ingredient in our proof are classification results on maximal-dimensional spaces of commutative matrices satisfying certain generating conditions. Our primary motivation is the study of the ind-scheme \[ \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{\infty}}^{\oplus r}) := \underset{n\rightarrow \infty}{colim} \mathrm{Quot}_d(\mathcal{O}_{\mathbb{A}^{n}}^{\oplus r}). \] Finally, we compute the cohomology (with integral coefficients) of the Quot scheme $\mathrm{Quot}_2(\mathcal{O}_{\mathbb{A}^n}^{\oplus r})$, confirming, in the case $d=2$, a conjecture of Pandharipande.