Bounding embedded singularities of Hilbert schemes of points on affine three space

Abstract

The Hilbert scheme [Formula: see text] of [Formula: see text] points on [Formula: see text] can be expressed as the critical locus of a regular function on a smooth variety [Formula: see text]. Recent development in birational geometry suggests a study of singularities of the pair [Formula: see text] using jet schemes. In this paper, we use a comparison between [Formula: see text] and the scheme [Formula: see text] of three commuting [Formula: see text] matrices to estimate the log canonical threshold of [Formula: see text]. As a consequence, we see that although both [Formula: see text] and [Formula: see text] have asymptotic growth [Formula: see text], the largest multiplicity of any points on [Formula: see text] has at most linear growth [Formula: see text].