Elementary proof of reducedness of Hilbert schemes of points in higher dimensions

TL;DR

This paper provides a straightforward proof that Hilbert schemes of points on nonsingular algebraic varieties are reduced in all dimensions, using a new criterion for algebra integrality.

Contribution

It introduces a simple proof of the reducedness of Hilbert schemes of points in higher dimensions based on a novel criterion for algebra integrality.

Findings

Hilbert schemes of points are reduced for all dimensions and lengths.

The proof relies on a new criterion for algebra integrality.

Simplifies previous complex proofs of reducedness.

Abstract

The criterion for an affine primary algebra over the field to be integral, is proven. Using this criterion we give a simple proof that Hilbert scheme of 0-dimensional subschemes of length $l$ of nonsingular $d$-dimensional algebraic variety is reduced for all $d$ and $l$.