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

Abstract

We give an Atiyah–Drinfel’d–Hitchin–Manin (ADHM) description of the Quot scheme of points [Formula: see text] of length [Formula: see text] and rank [Formula: see text] on affine spaces [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] and its punctual version [Formula: see text] where [Formula: see text] is a fixed point on a smooth affine variety [Formula: see text]. In this last case, we also study a connectedness result, for some special cases of higher [Formula: see text] and [Formula: see text].