On the universal family of Hilbert schemes of points on a surface

For a smooth quasi-projective surface X and an integer n≥3, we show that the universal family Zn over the Hilbert scheme Hilbn(X) of n points has non-Q-Gorenstein, rational singularities, and that the Samuel multiplicity μ at a closed point on Zn can be computed in terms of the dimension of the socle. We also show that μ≤n.