Resolution of singularities

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In algebraic geometry, the problem of resolution of singularities asks whether any 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 important open problem in dimensions at least 4.

Contents 1 Definitions 2 History 3 Resolution for schemes 4 External links 5 References

[edit] 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.

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.

[edit] 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), Kollar (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.

[edit] Resolution for schemes

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.

[edit] 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

[edit] References

• Abhyankar, Shreeram S. (1956), “Local uniformization on algebraic surfaces over ground fields of characteristic p≠0”, Ann. of Math.(2) 63: 491-526, <http://links.jstor.org/sici?sici=0003-486X%28195605%292%3A63%3A3%3C491%3ALUOASO%3E2.0.CO%3B2-1> 
• Abhyankar, Shreeram S. (1966), Resolution of singularities of embedded algebraic surfaces, Acad. Press, ISBN 3540637192  (1998 2nd edition)
• Abramovich, D, & de Jong, A. J. (1997), “Smoothness, semistability, and toroidal geometry.”, J. Algebraic Geom. 6 (4): 789-801, <http://arxiv.org/abs/alg-geom/9603018> 
• Albanese, G. (1924), “Transformazione birazionale di una superficie algebrica in un altra priva di punti multiple”, Rend. Circ. Mat. Palermo 48 
• Bierstone, Edward & Milman, Pierre D. (1997), “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant.”, Invent. Math. 128 (2): 207-302, DOI 10.1007/s002220050141 
• Bogomolov, Fedor A. & Pantev, Tony G. (1996), “Weak Hironaka theorem.”, Math. Res. Lett. 3 (3): 299-307, <http://arxiv.org/abs/alg-geom/9603019> 
• Chisini, O. (1921), “La risoluzione delle singolarita di una superficie”, Mem. Acad. Bologna 8 
• Cutkosky, Steven Dale (2004), Resolution of Singularities, ISBN 0821835556 
• Danilov, V.I. (2001), “Resolution of singularities”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• de Jong, A. J. (1996), “Smoothness, semi-stability and alterations.”, Inst. Hautes Études Sci. Publ. Math. 83: 51-93, <http://www.numdam.org/item?id=PMIHES_1996__83__51_0> 
• Encinas, S. & Hauser, Herwig (2002), “Strong resolution of singularities in characteristic zero.”, Comment. Math. Helv. 77 (4): 821-845, <http://arxiv.org/abs/math/0211423> 
• Encinas, S. & Villamayor, O. (1998), “Good points and constructive resolution of singularities.”, Acta Math. 181 (1): 109--158, DOI 10.1007/BF02392749 
• Grothendieck, A. & Dieudonne, J. (1965), “Eléments de géométrie algébrique”, Publ. Math. IHES 24, <http://www.numdam.org/item?id=PMIHES_1965__24__5_0> 
• Hauser, Herwig (2000), “Resolution of singularities 1860-1999.”, Resolution of singularities (Obergurgl, 1997), vol. 181, Progr. Math., Birkhäuser, pp. 5-36, ISBN 0817661786 
• Hauser, Herwig (2003), “The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand)”, Bull. Amer. Math. Soc. (N.S.) 40 (3): 323-403, <http://www.ams.org/bull/2003-40-03/S0273-0979-03-00982-0/home.html> 
• Hironaka, Heisuke (1964), “Resolution of singularities of an algebraic variety over a field of characteristic zero. I”, Ann. of Math. (2) 79: 109-203, <http://links.jstor.org/sici?sici=0003-486X%28196401%292%3A79%3A1%3C109%3AROSOAA%3E2.0.CO%3B2-M>  and part II, pp. 205-326, <http://links.jstor.org/sici?sici=0003-486X%28196403%292%3A79%3A2%3C205%3AROSOAA%3E2.0.CO%3B2-I> 
• Kollar, Janos (2007), Lectures on Resolution of Singularities, ISBN 0691129231  (similar to his Resolution of Singularities -- Seattle Lecture.
• Levi, B. (1899), “Risoluzione delle singolarita puntualli delle superficie algebriche”, Atti. Acad. Torino 34 
• Lipman, J. (1978), “Desingularization of two-dimensional schemes”, Ann. Math. 107: 151-207, <http://links.jstor.org/sici?sici=0003-486X%28197803%292%3A107%3A2%3C151%3ADOTS%3E2.0.CO%3B2-Y> 
• Walker, Robert J. (1935), “Reduction of the Singularities of an Algebraic Surface”, The Annals of Mathematics 2nd Ser. 36 (2): 336-365, <http://links.jstor.org/sici?sici=0003-486X%28193504%292%3A36%3A2%3C336%3AROTSOA%3E2.0.CO%3B2-T> 
• Wlodarczyk, Jaroslaw (2005), “Simple Hironaka resolution in characteristic zero.”, J. Amer. Math. Soc. 18 (4): 779-822, <http://www.ams.org/jams/2005-18-04/S0894-0347-05-00493-5/home.html> 
• Zariski, Oscar (1939), “The reduction of the singularities of an algebraic surface.”, Ann. of Math. (2) 40: 639-689, <http://links.jstor.org/sici?sici=0003-486X%28193907%292%3A40%3A3%3C639%3ATROTSO%3E2.0.CO%3B2-N> 
• Zariski, Oscar (1944), “Reduction of the singularities of algebraic three dimensional varieties.”, Ann. of Math. (2) 45: 472-542, <http://links.jstor.org/sici?sici=0003-486X%28194407%292%3A45%3A3%3C472%3AROTSOA%3E2.0.CO%3B2-A>