Galois closures and elementary components of Hilbert schemes of points

TL;DR

This paper generalizes Galois closures for ring extensions and uses them to identify new elementary components of Hilbert schemes of points, revealing their structure and parametrization.

Contribution

It introduces a generalized Galois closure operation and constructs an infinite family of elementary components of Hilbert schemes of points.

Findings

Constructed new elementary components parametrizing algebras supported at a point

Produced secondary families of elementary components via Galois closures and socle elements

Extended the functorial Galois closure concept to broader algebraic contexts

Abstract

Bhargava and the first-named author of this paper introduced a functorial Galois closure operation for finite-rank ring extensions, generalizing constructions of Grothendieck and Katz--Mazur. In this paper, we generalize Galois closures and apply them to construct a new infinite family of irreducible components of Hilbert schemes of points. We show that these components are elementary, in the sense that they parametrize algebras supported at a point. Furthermore, we produce secondary families of elementary components obtained from Galois closures by modding out by suitable socle elements.