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

The Hilbert scheme Hilbⁿ C³ 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, Hilbⁿ C³) using jet schemes. In this paper, we use a comparison between Hilbn C³ and the scheme C³,n of three commuting n × n matrices to estimate the log canonical threshold of (X, Hilbn C³). As a consequence, we see that although both dimX and dim Hilbn C³ have asymptotic growth O(n²), the largest multiplicity of any points on Hilbn C³ has at most linear growth O(n).