Explicit projective embeddings of standard opens of the Hilbert scheme of points

TL;DR

This paper provides explicit descriptions of how standard open subsets of the Hilbert scheme of points are embedded into Grassmannians, clarifying their geometric structure through intersections with open affines and Fitting ideals.

Contribution

It offers a detailed, explicit construction of embeddings of standard opens of the Hilbert scheme into Grassmannians, enhancing understanding of their geometric and algebraic properties.

Findings

Explicit embeddings of standard opens into Grassmannians

Characterization of these opens via Fitting ideals

Clarification of the geometric structure of the Hilbert scheme

Abstract

We describe explicitly how certain standard opens of the Hilbert scheme of points are embedded into Grassmannians. The standard opens of the Hilbert scheme that we consider are given as the intersection of a corresponding basic open affine of the Grassmannian and a closed stratum determined by a Fitting ideal.