A note on the ADHM description of Quot schemes of points on affine spaces

TL;DR

This paper extends the ADHM description to Quot schemes of points on affine spaces, providing new insights into their geometric properties like irreducibility, reducedness, and connectedness in certain cases.

Contribution

It introduces a unified ADHM framework for Quot schemes of points on affine spaces, generalizing previous models and establishing new geometric properties.

Findings

Proves irreducibility and reducedness of Quot schemes in certain cases.

Establishes connectedness results for specific parameters.

Extends ADHM description to higher rank and length cases.

Abstract

We give an ADHM description of the Quot scheme of points ${\rm Quot}_{\mathbb{C}^{n}}(c,r),$ of length $c$ and rank $r$ on affine spaces $\mathbb{C}^{n}$ which naturally extends both Baranovsky's representation of the punctual Quot scheme on a smooth surface and the Hilbert scheme of points on affine spaces $\mathbb{C}^{n},$ described by the first author and M. Jardim. Using results on the variety of commuting matrices, and combining them with our construction, we prove new properties concerning irreducibility and reducedness of ${\rm Quot}_{\mathbb{C}^{n}}(c,r)$ and its punctual version ${\rm Quot}_{\mathbb{Y}}^{[p]}(c,r),$ where $p$ is a fixed point on a smooth affine variety $\mathbb{Y}.$ In this last case we also study a connectedness result, for some special cases of higher $r$ and $c.$