Generic fibers of the generalized Springer resolution of type \(A\) (Q557593)

Let \(G\) be a semisimple (connected) complex algebraic group with Lie algebra \(\mathfrak{g}\), on which \(G\) acts by the adjoint action, and let \(P\) be a standard parabolic subgroup of \(G\) with Lie algebra \(\mathfrak{p}\). There is a unique nilpotent \(G\)-orbit \(\mathcal{O}_{\mathfrak{p}}\) such that the set \(\mathcal{O}_{\mathfrak{p}} \cap \mathfrak{n}_{\mathfrak{p}}\) is open and dense in \(\mathfrak{n}_{\mathfrak{p}}\), the nilradical of \(\mathfrak{p}\), and is called the Richardson orbit associated with \(\mathfrak{p}\). Let \(G \times ^P\mathfrak{n}_{\mathfrak{p}}\) be the quotient space of \(G \times \mathfrak{n}_{\mathfrak{p}}\) by the right action of \(P\) given by \((g,x).p = (gp, p^{-1}.x)\), where \(g \in G\), \(x \in \mathfrak{n}_{\mathfrak{p}}\), and \(p \in P\). The mapping \(f_{\mathfrak{p}}: G \times ^P\mathfrak{n}_{\mathfrak{p}} \rightarrow \mathfrak{g}\) defined by \(g \ast x \mapsto g.x\), where \(g \ast x \in G \times ^P\mathfrak{n}_{\mathfrak{p}}\) is the equivalence class of \((g,x)\), is called the generalized Springer resolution. After the proof of a general result on some irreducible components of the fibers of \(f_{\mathfrak{p}}\), the author investigates the generalized Springer resolution when \(G = \mathbf{SL}(n,\mathbb{C})\). He shows that, in this case, the generalized Springer fiber \(f_{\mathfrak{p}}^{-1}(x)\) is isomorphic either to a Dynkin curve or to a projective space for all \(x \in \mathcal{O} \cap \mathfrak{n}_{\mathfrak{p}}\), where \(\mathcal{O}\) is any nilpotent \(G\)-orbit included in the closure of \(\mathcal{O}_{\mathfrak{p}}\). Then, he applies results of [\textit{H. Esnault}, Singularités rationelles et groupes algébriques, Thèse de 3ème cycle, Paris VII (1976)] to prove that, in some cases, the generalized Springer resolution restricts to the minimal resolution of a normal surface with a rational double point of type \(A_r\), for a well-defined \(r\).