Hilbert schemes with two Borel-fixed points

TL;DR

This paper characterizes certain Hilbert schemes with exactly two Borel-fixed points, analyzing their structure, smoothness, and singularities, and provides examples with three Borel-fixed points.

Contribution

It provides a complete characterization of Hilbert schemes with two Borel-fixed points, including their smoothness, irreducible components, and singularity structure.

Findings

Hilbert schemes with two Borel-fixed points are reduced and have at most two irreducible components.

Irreducible components are Cohen-Macaulay and normal near Borel-fixed points.

The paper presents numerous examples of Hilbert schemes with three Borel-fixed points.

Abstract

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or their irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points.