On Birational Transformations of Hilbert Schemes of an Algebraic Surface

We give a precise description of the closure $\Gamma_f$ of the graph of the birational isomorphism $f\colon\operatorname{Hilb}^d\widetilde S\dasharrow\operatorname{Hilb}^dS$ of the Hilbert schemes of points on algebraic surfaces that corresponds to the blow-up $\sigma\colon\widetilde S\to S$ centered at a point on the smooth algebraic surface $S$. We prove that the projection $\operatorname{pr}_{\widetilde H}\colon\Gamma_f\to\widetilde H=\operatorname{Hilb}^d\widetilde S$ 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 $d\le2$, and a precise description of the decomposition of the second projection $\operatorname{pr}_H\colon\Gamma_f\to H=\operatorname{Hilb}^dS$ into a composition of two blow-ups with smooth centers in the nontrivial case $d=2$ is given.