The cotangent space at a monomial ideal of the Hilbert scheme of points of an affine space

TL;DR

This paper investigates the cotangent space at monomial ideals in the Hilbert scheme of points in affine space, providing explicit bases, conditions for smoothness, and characterizations for specific cases.

Contribution

It constructs explicit bases for cotangent spaces at monomial ideals and characterizes smoothness conditions for these ideals in low-dimensional cases.

Findings

Explicit basis construction for cotangent spaces at monomial ideals.

Conditions for when the basis is or isn't a true basis.

Characterization of smooth points for r=2 and r=3 variables.

Abstract

Let k be an algebraically closed field. We study the cotangent space of a point t corresponding to a monomial ideal I of k[x_1, ..., x_r] in the Hilbert scheme of n points of affine r-space (so the k-dimension of k[x_1, ..., x_r]/I = colength of I = n). Since t lies in the closure of the locus corresponding to subschemes supported at n distinct points of A^r_k, one knows that the k-dimension of the cotangent space is always >= r*n, and that t is nonsingular if and only if the dimension equals r*n. We construct an explicit linearly independent set S of cotangent vectors of size r*n, and then explore conditions on I under which S either is or is not a basis of the cotangent space. In particular, we give a condition on I sufficient for S to be a basis (equivalently, for t to be nonsingular) that holds for every monomial ideal in the case of r = 2 variables, and that characterizes such ideals when r = 3. We also give an easily-checked condition on I sufficient for S not to be a basis.