Counting Dimensions of Tangent Spaces to Hilbert Schemes of Points

TL;DR

This paper advances understanding of smoothness in Hilbert schemes of points by analyzing tangent space dimensions through Young diagram combinatorics, providing new criteria for smoothness in 2D and 3D cases.

Contribution

It introduces new combinatorial conditions based on Young diagrams that determine smooth points on Hilbert schemes of points in 2 and 3 dimensions.

Findings

Rectangular regions between Young diagrams imply smoothness.

Horizontal rectangular layers in 3D diagrams imply smoothness.

Provides sufficient conditions for smoothness in nested schemes.

Abstract

In this paper we prove two results which further classify smoothness properties of Hilbert schemes of points. This is done by counting classes of arrows on Young diagrams corresponding to monomial ideals, building on the approach taken by Jan Cheah to show smoothness in the 2 dimensional case. We prove sufficient conditions for points to be smooth on Hilbert schemes of points in 3 dimensions and on nested schemes in 2 dimensions in terms of the geometry of the Young diagram. In particular, we proved that when the region between the two diagrams at a point of the nested scheme is rectangular, the corresponding point is smooth. We also proved that if the three dimensional Young diagram at a point can be oriented such that its horizontal layers are rectangular, then the point is smooth.