A monotonicity conjecture for the local maximal singularity of the Hilbert scheme of points

TL;DR

This paper explores conjectures related to the maximal singularities of the Hilbert scheme of points, proposing a monotonicity conjecture that links tangent space dimensions to ideal exponents.

Contribution

It introduces a new monotonicity conjecture and provides a conjectural sufficient condition for maximal singularities in the Hilbert scheme.

Findings

Proposes a monotonicity conjecture for tangent space dimensions.

Provides a conjectural sufficient condition for maximal singularities.

Links ideal exponents to the maximal dimension of tangent spaces.

Abstract

The Brian\c{c}on-Iarrobino conjecture predicts the maximum singularity of the Hilbert scheme of a tetrahedral number of points. As for the maximal singularities of the Hilbert scheme of a non-tetrahedral number of points, the second named author gave some separate conjectural necessary and sufficient conditions. In this paper, we provide a conjectural sufficient condition for the necessary condition, and propose a monotonicity conjecture which predicts that for a fixed colength $l$, the maximal dimension of the tangent space over all the Borel-fixed ideals of colength $l$ is increasing with respect to the smallest pure exponent of the ideal.