Method of Proof in Characteristic Zero
There are many constructions of strong desingularization but all of them give essentially the same result. In every case the global object (the variety to be desingularized) is replaced by local data (the ideal sheaf of the variety and those of the exceptional divisors and some orders that represents how much should be resolved the ideal in that step). With this local data the centers of blowing-up are defined. The centers will be defined locally and therefore it is a problem to guarantee that they will match up into a global center. This can be done by defining what blowings-up are allowed to resolve each ideal. Done this appropriately will make the centers match automatically. Another way is to define a local invariant depending on the variety and the history of the resolution (the previous local centers) so that the centers consist of the maximum locus of the invariant. The definition of this is made such that making this choice is meaningful, giving smooth centers transversal to the exceptional divisors.
In either case the problem is reduced to resolve singularities of the tuple formed by the ideal sheaf and the extra data (the exceptional divisors and the order, d, to which the resolution should go for that ideal). This tuple is called a marked ideal and the set of points in which the order of the ideal is larger than d is called its co-support. The proof that there is a resolution for the marked ideals is done by induction on dimension. The induction breaks in two steps:
- Functorial desingularization of marked ideal of dimension n − 1 implies functorial desingularization of marked ideals of maximal order of dimension n.
- Functorial desingularization of marked ideals of maximal order of dimension n implies functorial desingularization of (a general) marked ideal of dimension n.
Here we say that a marked ideal is of maximal order if at some point of its co-support the order of the ideal is equal to d. A key ingredient in the strong resolution is the use of the Hilbert–Samuel function of the local rings of the points in the variety. This is one of the components of the resolution invariant.
Read more about this topic: Resolution Of Singularities
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