The singularities of the principal component of the Hilbert scheme of points

Haiman proved the remarkable result that the isospectral Hilbert scheme of points in the plane is normal, Cohen–Macaulay and Gorenstein. This implies the n! conjecture and the positivity conjecture for the Kostka–Macdonald coefficients. In addition, he conjectured that the isospectral Hilbert scheme over the principal component of Hilbn(Cd) is Cohen–Macaulay for any d,n⩾1. We provide a counterexample to the conjecture.