On Birational Transformations of Hilbert Schemes of an Algebraic Surface

Abstract

We give a precise description of the closure $\Gamma _f$ of the graph of the birational isomorphism $f:Hilb^d \widetilde S - - \to Hilb^d S$ of the Hilbert schemes of points on algebraic surfaces that corresponds to the blow-up $\sigma :\widetilde S \to S$ centered at a point on the smooth algebraic surface $S$. We prove that the projection $pr_{\widetilde H} :\Gamma _f \to {\widetilde H} = Hilb^d \widetilde S$ is the blow-up centered in the incidence subvariety $R \subset \widetilde H$ that parametrizes $d$-tuples of points in $\widetilde S$ such that at least two of these points are incident to the exceptional line of the blow-up $\sigma$; here $R$ is endowed with a scheme structure by means of a suitable sheaf of Fitting ideals. It is shown that $\Gamma _f$ is smooth only for $d \leqslant 2$, and a precise description of the decomposition of the second projection ${\text{pr}}_H :\Gamma _f \to H = {\text{Hilb}}^d S$ into a composition of two blow-ups with smooth centers in the nontrivial case $d = 2$ is given.