Patching local uniformizations.

*(English)*Zbl 0782.14009To make a desingularization of a scheme \(X\), the classical way now is to define an upper-semicontinuous function \(\Phi\) defined on the singular locus of \(X\), with value in an ordered set \(E\). \(\Phi\) needs to have three properties:

(1) The set \(Y\) of points where \(\Phi\) is maximal is regular.

(2) If you blow-up \(Y\) on \(X'\), the strict transform of \(X\), \[ \sup\bigl\{\Phi(x'):x'\in X'\bigr\}<{\sup\bigl\{\Phi(x):x\in X\bigr\}}=\sup\bigl\{\Phi(x):x\in Y\bigr\}. \] (3) There is no infinite strictly decreasing sequence of elements of \(E\).

That is what the author did in a previous article [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], in the case of characteristic 0. There were still some open questions:

\(Q_ 1\): Does the algorithm commute under smooth morphisms?

\(Q_ 2\): Can he lift on his desingularization of \(X\) any action of group of isomorphisms acting on \(X\)?

In the article under review, the author introduces the notions of trees, which are concatenations of blowing-ups and smooth morphisms, and of groves which are approximatively “sheaves of trees”. Groves generalize the idealistic exponents used to make desingularization. The main result in this paper is that the function \(\Phi\) used in the previous paper cited above can be defined only with the set of groves of \(X\) which are geometrically constructed to control the composition of blowing-ups and smooth morphisms. This is a very strong result: it implies that the function \(\Phi\) is canonical and commutes under any smooth morphism, i.e., if \(f:X_ 1\to X_ 2\) is a smooth morphism, for any point \(x\in\text{Sing}(X_ 1)\), \(\Phi(x)=\Phi\bigl(f(x)\bigr)\). As a corollary, the author’s answers \(Q_ 1\) and \(Q_ 2\).

At the end of the paper, two examples are completely exposed.

(1) The set \(Y\) of points where \(\Phi\) is maximal is regular.

(2) If you blow-up \(Y\) on \(X'\), the strict transform of \(X\), \[ \sup\bigl\{\Phi(x'):x'\in X'\bigr\}<{\sup\bigl\{\Phi(x):x\in X\bigr\}}=\sup\bigl\{\Phi(x):x\in Y\bigr\}. \] (3) There is no infinite strictly decreasing sequence of elements of \(E\).

That is what the author did in a previous article [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 1-32 (1989; Zbl 0675.14003)], in the case of characteristic 0. There were still some open questions:

\(Q_ 1\): Does the algorithm commute under smooth morphisms?

\(Q_ 2\): Can he lift on his desingularization of \(X\) any action of group of isomorphisms acting on \(X\)?

In the article under review, the author introduces the notions of trees, which are concatenations of blowing-ups and smooth morphisms, and of groves which are approximatively “sheaves of trees”. Groves generalize the idealistic exponents used to make desingularization. The main result in this paper is that the function \(\Phi\) used in the previous paper cited above can be defined only with the set of groves of \(X\) which are geometrically constructed to control the composition of blowing-ups and smooth morphisms. This is a very strong result: it implies that the function \(\Phi\) is canonical and commutes under any smooth morphism, i.e., if \(f:X_ 1\to X_ 2\) is a smooth morphism, for any point \(x\in\text{Sing}(X_ 1)\), \(\Phi(x)=\Phi\bigl(f(x)\bigr)\). As a corollary, the author’s answers \(Q_ 1\) and \(Q_ 2\).

At the end of the paper, two examples are completely exposed.

Reviewer: V.Cossart (Versailles)

##### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |