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

Abstract

The Hilbert scheme ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ of $n$ points on ${ℂ}^3$ can be expressed as the critical locus of a regular function on a smooth variety ${𝒳}$. Recent development in birational geometry suggests a study of singularities of the pair $({𝒳},{\mathrm{\text{Hilb}}}^n{ℂ}^3)$ using jet schemes. In this paper, we use a comparison between ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ and the scheme $C_{3,n}$ of three commuting $n\times n$ matrices to estimate the log canonical threshold of $({𝒳},{\mathrm{\text{Hilb}}}^n{ℂ}^3)$. As a consequence, we see that although both $dim{𝒳}$ and $dim{\mathrm{\text{Hilb}}}^n{ℂ}^3$ have asymptotic growth $O(n^2)$, the largest multiplicity of any points on ${\mathrm{\text{Hilb}}}^n{ℂ}^3$ has at most linear growth $O(n)$.