Resolution of singularities: Wikis

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Strong desingularization of X: = (x2y3 = 0)W: =2. Observe that the resolution does not stop after the first blowing-up, when the strict transform is smooth, but when it is simple normal crossings with the exceptional divisors.

In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety has a non-singular model (a non-singular variety birational to it). For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic p it is an open problem in dimensions at least 4.

Contents

Definitions

A variety over a field has a weak resolution of singularities if we can find a complete non-singular variety birational to it, in other words with the same function field. In practice it is convenient to ask for a stronger condition as follows: a variety X has a resolution of singularities if we can find a non-singular variety X′ and a proper birational map from X′ to X, which is an isomorphism over the non-singular points of X. (The condition that the map is proper is needed to exclude trivial solutions, such as taking X′ to be the subvariety of non-singular points of X.)

More generally, it is often useful to resolve the singularities of a variety X embedded into a larger variety W. Suppose we have a closed embedding of X into a regular variety W. A strong desingularization of X is given by a proper birational morphism from a regular variety W′ to W subject to some of the following conditions (the exact choice of conditions depends on the author):

  1. The strict transform X′ of X is regular, and transverse to the exceptional locus of the blowup (so in particular it resolves the singularities of X).
  2. W′ is constructed by repeatedly blowing up regular closed subvarieties, transverse to the exceptional locus of the blowup.
  3. The construction of W′ is functorial for smooth morphisms to W and embeddings of W into a larger variety. (It cannot be made functorial for all non-smooth morphisms in any reasonable way.)
  4. The morphism from X′ to X does not depend on the embedding of X in W. Or in general, the sequence of blowings up is functorial with respect to smooth morphisms.

Hironaka showed that there is a strong desingularization satisfying the first two conditions above whenever X is defined over a field of characteristic 0, and his construction was improved by several authors (see below) so that it satisfies all four conditions above.

History

Resolution of singularities of curves is easy and was well known in the 19th century. There are many ways of proving it; the two most common are to repeatedly blow up singular points, or to take the normalization of the curve. Normalization removes all singularities in codimension 1, so it works for curves but not in higher dimensions.

Resolution for surfaces over the complex numbers was given informal proofs by Levi (1899), Chisini (1921) and Albanese (1924). A rigorous proof was first given by Walker (1935), and an algebraic proof for all fields of characteristic 0 was given by Zariski (1939). Abhyankar (1956) gave a proof for surfaces of non-zero characteristic. Resolution of singularities has also been shown for all excellent 2-dimensional schemes (including all arithmetic surfaces) by Lipman (1978). The usual method of resolution of singularities for surfaces is to repeatedly alternate normalizing the surface (which kills codimension 1 singularities) with blowing up points (which makes codimension 2 singularities better, but may introduce new codimension 1 singularities).

For 3-folds the resolution of singularities was proved in characteristic 0 by Zariski (1944), and in characteristic greater than 5 by Abhyankar (1966).

Resolution of singularities in characteristic 0 in all dimensions was first proved by Hironaka (1964). He proved that it was possible to resolve singularities of varieties over fields of characteristic 0 by repeatedly blowing up along non-singular subvarieties, using a very complicated argument by induction on the dimension. Simplified versions of his formidable proof were given by several people, including Bierstone & Milman (1997), Encinas & Villamayor (1998), Encinas & Hauser (2002), Cutkosky (2004), Wlodarczyk (2005), Kollár (2007). Some of the recent proofs are about a tenth of the length of Hironaka's original proof, and are easy enough to give in an introductory graduate course. For an expository account of the theorem, see (Hauser 2003) and for a historical discussion see (Hauser 2000).

de Jong (1996) found a different approach to resolution of singularities, which was used by Bogomolov & Pantev (1996) and by Abramovich & de Jong (1997) to prove resolution of singularities in characteristic 0. De Jong's method gave a weaker result for varieties of all dimensions in characteristic p, which was strong enough to act as a substitute for resolution for many purposes. De Jong proved that for any variety X over a field there is a dominant proper morphism which preserves the dimension from a regular variety onto X. This need not be a birational map, so is not a resolution of singularities, as it may be generically finite to one and so involves a finite extension of the function field of X. De Jong's idea was to try to represent X as a fibration over a smaller space Y with fibers that are curves (this may involve modifying X), then eliminate the singularities of Y by induction on the dimension, then eliminate the singularities in the fibers.

Resolution for schemes and status of the problem

It is easy to extend the definition of resolution to all schemes. Not all schemes have resolutions of their singularities: Grothendieck (1965, section 7.9) showed that if a locally Noetherian scheme X has the property that one can resolve the singularities of any finite integral scheme over X, then X must be quasi-excellent. Grothendieck also suggested that the converse might hold: in other words, if a locally Noetherian scheme X is reduced and quasi excellent, then it is possible to resolve its singularities. When X is defined over a field of characteristic 0, this follows from Hironaka's theorem. In general it would follow if it is possible to resolve the singularities of all integral complete local rings.

Writing towards the end of the 1960s, Grothendieck identified two foundational problems in algebraic geometry as having the greatest urgency: resolution was one, the other being the standard conjectures.[1]

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:

  1. Functorial desingularization of marked ideal of dimension n − 1 implies functorial desingularization of marked ideals of maximal order of dimension n.
  2. 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.

Other variants of resolutions of singularities

  • It happens that after the resolution the total transform, the union of the strict transform, X, and the exceptional divisors, is a variety with singularities of the simple normal crossings type. Then it is natural to consider the possibility of resolving singularities without resolving this type of singularities, this is finding a resolution that is an isomorphism over the set of smooth and simple normal crossing points. When X is a divisor, i.e. it can be embedded as a codimension one subvariety in a smooth variety it is known to be true the existence of the strong resolution avoiding simple normal crossing points. The general case or generalizations to avoid different types of singularities are still not known.
  • Some cases are known to be not true. For example, it is not possible to resolve singularities avoiding blowing-up the normal crossings singularities. In fact, to resolve the pinch point singularity the whole singular locus needs to be blown up, including points where normal crossing singularities are present. Also it is not possible to get a strong resolution functorial with respect to every morphism.

References

Notes

  1. ^ [1] (last sentence of paper, p.198).

External links

  • Some pictures of singularities and their resolutions
  • SINGULAR: a computer algebra system with packages for resolving singularities.
  • Notes and lectures for the Working Week on Resolution of Singularities Tirol 1997, September 7-14, 1997, Obergurgl, Tirol, Austria
  • Lecture notes from the Summer School on Resolution of Singularities, June 2006, Trieste, Italy.
  • desing - A computer program for resolution of singularities

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