Resolving singularities with Cartan's prolongation (Q948491)

Let \(X\) be a purely \(d\)-dimensional algebraic (or complex analytic) subvariety of \({\mathbb{A}}^n\), with singular locus \(S\). The \textit{prolongation} (or Nash blowing-up, or Nash modification) \(X_1\) of \(X\) is the closure of the graph of the morphism \(X \;S \to \text{Grass}(d,n)\) sending \(x \in X \setminus S\) to the tangent space \(T_{X,x}\). There is a natural projection \(\pi:X_1 \to X\) which is an isomorphism away from \(S\). This construction easily generalizes to general equidimensional varieties. It is known that (in characteristic zero) \(\pi\) is an isomorphism if and only if \(X\) is regular. From this it easily follows that if \(X\) is a curve a finite sequence of prolongations desingularizes \(X\). In this paper, a finer version of resolution of singularities for a plane curve \(C\) is studied. Namely, the authors look for an analogue of the classical theorem on ``embedded resolution'', that says that by means of a finite sequence of blowing-ups whose centres are points we obtain a morphism \(f:F \to {\mathbb{A}}^2\) such that the reduced inverse image of \(C\) is a normal crossings divisor. Also a graph \(\Gamma\) whose vertices correspond to the irreducible components of the exceptional locus of \(f\) is introduced; it determines the structure of the singularities of \(C\). An initial difficulty when using prolongations is the fact that the prolongation of a hypersurface is not a hypersurface of any reasonable smooth ambient space. But, if \(C\) is a plane curve (in \({\mathbb{C}}^2\)), the authors show that the successive prolongations \(C_1, C_2, \ldots \) of \(C\) essentially remain in the planar situation. Namely, for any integer \(r>0\), there is a sequence of varieties and morphisms: \[ {\mathbb{C}}^2 \leftarrow {\mathbb{P}}^1({\mathbb{C}}^2) \leftarrow \cdots \leftarrow \mathbb{P}^r({\mathbb{C}}^2) \] such that each iterated prolongation \(C_j\) of \(C\) is a curve in \(\mathbb{P}^1({\mathbb{C}}^2)\), \(\mathbb{P}^{j+1}({\mathbb{C}}^2)\) is a \({\mathbb{P}}^1\)-bundle over \(\mathbb{P}^j({\mathbb{C}}^2)\) and, moreover, on each \(\mathbb{P}^j({\mathbb{C}}^2)\) there is a rank-2 complex distribution \({\Delta}_j\) such that \(C_j\) is tangent to \(\Delta _j\) (for all \(j\)). A curve in \(\mathbb{P}^j({\mathbb{C}}^2)\), tangent to \(\Delta _j\) and mapping to a point by the natural composition morphism \(\mathbb{P}^j({\mathbb{C}}^2) \to {\mathbb{C}}^2\) is called a \textit{critical} curve. They play a role analogous to the exceptional curves in the classical case (i.e., when we work with blowing-ups of points). Precisely, the authors prove that, given a germ \((C,0) \subset ({\mathbb{C}}^2,0)\) of an analytically irreducible curve (a branch): (a) For an index \(r\), in the sequence above the reduced inverse image of \(C\) in \(\mathbb{P}^r({\mathbb{C}}^2)\) is of the form \(D=C_r \cup V_1 \cup \ldots \cup V_r\), where \(C_r\) is the iterated prolongation of \(C\), \(V_1, \ldots, V_r\) are the critical curves, each pair of irreducible components of \(D\) intersect transversally (relative to the distribution \(\Delta _r\)) and any three of them have empty intersection.; (b) a graph \(\Gamma '\), whose vertices correspond to \(V_1, \ldots, V_r\), can be constructed, and this graph is isomorphic to the graph \(\Gamma\) of the classical situation already mentioned. To determine the edges of \(\Gamma '\) the authors introduce a relation of \textit{incidence}, different form that used for the edges of \(\Gamma\) of the classical case. The proof of (b) is hard. It requires the introduction of several auxiliary notions, specially the \textit{RVT codes} (related to Puiseux characteristics), and delicate computations. In a concluding section the authors discuss some possible extensions and open problems.