Functoriality in resolution of singularities
The authors investigate a procedure to resolve singularities that has reasonable ``functorial'' properties. The main result is expressed in terms of ``marked ideals'', that is a 5-tuple \(\mathcal I = (M,N,E,I,d)\) where \(N\) is a subvariety of \(M\), both smooth over a field of characteristic zero, \(I\) is a coherent sheaf of ideals of \(N\), \(E\) is a normal crossings divisor of \(M\), transversal to \(N\) and \(d\) is a positive integer. The singular locus, or cosupport, of \(\mathcal I\) is the set of points \(x\) of \(N\) such that the order of the stalk \(I_x\) is \(\geq d\). One may define the transformation of such a marked ideal with center a suitable smooth subvariety of \(N\) (the \textit{admissible} transformations), the result is a new marked ideal. The objective is to obtain, by means of a finite sequence of admissible transformations, a marked ideal whose cosupport is empty. Such a sequence is called a resolution sequence. If this can be done in a reasonable constructive (or algorithmic) way other more classical desingularization theorems follow rather easily. (This is explained in the present article). The authors define a notion of equivalence of marked ideals as follows. A test transformation is either an admissible one, or one determined by the blowing-up of a center which is the intersection of two components of \(E\), or one induced by a projection \(M \times {\mathbb{A}}^1 \to M\). Two marked ideals \(\mathcal I = (M,N,E,I,d)\) and \(\mathcal J = (M,N',E,J,d')\) are equivalent if they have the same sequences of of test transformations. Then they prove that there is way to associate to each marked ideal a resolution sequence such that if \(\mathcal I\) and \(\mathcal J\) are equivalent, then the resolution sequence associated to \(\mathcal I\) is obtained by using the same centers as were needed for the sequence of \(\mathcal J\). Moreover, this procedure is compatible with smooth morphisms \(M' \to M\). This is what is meant by functoriality of the process. The procedure is a variant of that introduced by the authors in their fundamental paper \textit{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 128, No. 2, 207--302 (1997; Zbl 0896.14006)] incorporating some ideas from \textit{J. Włodarczyck}'s article [J. Am. Math. Soc. 18, No. 4, 779--822 (2005; Zbl 1084.14018)]. The exploitation of the explicit requirements on functoriality simplifies the verification of the fact that certain constructions are independent of the elections made, which is a hard problem in this type of desingularization work. The authors use the functorial character of the algorithm to show that it coincides with others recently introduced by J. Wlodarczyk and J. Kollár.
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