On birational transformations of Hilbert schemes of an algebraic surface (Q1810307)

Let \(S\) be a smooth algebraic surface over an algebraically closed field, \(\sigma : \widetilde S\rightarrow S\) the blow-up of \(S\) at a point \(x_0\) and \(l_0 = {\sigma}^{-1}(x_0)\) the exceptional divisor. Let \(H\) (resp., \(\widetilde H\)) denote the Hilbert scheme of 0-dimensional subschemes of \(S\) (resp., \(\widetilde S\)) of length \(d\). \(\sigma \) induces a birational map \(f : \widetilde H\dashrightarrow H\). The author shows that there exists a closed subscheme \(R\) of \(\widetilde H\) (defined by a suitable Fitting ideal sheaf) with support \(\{ [Z]\in \widetilde H \mid \text{length}(Z\cap l_0)\geq 2\} \) such that, if \(\widehat H\) is the blow-up of \(\widetilde H\) along \(R\), \(f\) can be extended to a \textit{morphism} \(\widehat f : \widehat H\rightarrow H\). If \(d = 2\) then \(R\) is smooth (in fact, \(R\simeq S^2l_0\simeq {\mathbb P}^2\)) hence \(\widetilde H\) is smooth, and \(\widehat f\) is the composition of two blow-ups with smooth centres. The first one is the blow-up \(\tau : H^{\prime}\rightarrow H\) of \(H\) along \(Q = \{ [Z]\in H\mid x_0\in Z\} \simeq \widetilde S\) and the second one is the blow-up of \(H^{\prime}\) along the unique section \(m_0\) with self-intersection -3 of the rational ruled surface \({\tau}^{-1}(l_0)\simeq {\mathbb F}_3\). For \(d\geq 3\), however, \(\widehat H\) is singular.