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Strong desingularization of
X: = (x^{2} −
y^{3} = 0)⊂
W: = ℝ
^{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.
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):
- 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).
- W′ is constructed by repeatedly blowing up regular
closed subvarieties, transverse to the exceptional locus of the
blowup.
- 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.)
- 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:
- 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.
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
- Abhyankar, Shreeram
S. (1956), "Local uniformization on
algebraic surfaces over ground fields of characteristic
p≠0", Ann. Of Math.(2) 63:
491–526, doi:10.2307/1970014,
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
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(1924), "Trasformazione birazionale di una superficie algebrica in
un'altra priva di punti multipli", Rend. Circ. Mat.
Palermo 48: 321, doi:10.1007/BF03014708
- 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,
Providence, RI: American Math. Soc., ISBN
0821835556
- Danilov, V.I.
(2001), "Resolution of
singularities", in Hazewinkel, Michiel, Encyclopaedia of
Mathematics, Kluwer Academic Publishers, ISBN
978-1556080104, http://eom.springer.de/R/r081580.htm
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(1996), "Smoothness, semi-stability
and alterations.", Inst. Hautes Études Sci. Publ.
Math. 83: 51–93, doi:10.1007/BF02698644, http://www.numdam.org/item?id=PMIHES_1996__83__51_0
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S.; Hauser, Herwig (2002), "Strong resolution of singularities in
characteristic zero.", Comment. Math. Helv.
77 (4): 821–845, doi:10.1007/PL00012443, http://arxiv.org/abs/math/0211423
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(1998), "Good points and constructive resolution of
singularities.", Acta Math. 181 (1):
109--158, doi:10.1007/BF02392749
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Dieudonné, 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), Progr. Math.,
181, 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, doi:10.1090/S0273-0979-03-00982-0, 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, doi:10.2307/1970486,
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
- Kollár,
János (2007), Lectures on Resolution of Singularities,
Princeton: Princeton University Press, 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, doi:10.2307/1971141,
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, doi:10.2307/1968575,
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, doi:10.1090/S0894-0347-05-00493-5, 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, doi:10.2307/1968949,
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, doi:10.2307/1969189,
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Notes
- ^
[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