In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety W with a proper birational map W→V. For varieties over fields of characteristic 0, this was proved by Heisuke Hironaka in 1964; while for varieties of dimension at least 4 over fields of characteristic p, it is an open problem.
Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety X, in other words a complete non-singular variety X′ with the same function field. In practice it is more convenient to ask for a different 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. 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 resolution morphism (so in particular it resolves the singularities of X). The map from the strict transform of X′ to X is an isomorphism away from the singular points of X. W′ is constructed by repeatedly blowing up regular closed subvarieties of W or more strongly regular subvarieties of X, transverse to the exceptional locus of the previous blowings up. 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 (not necessarily 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 three 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 conditions above.
Resolution of singularities of curves
Every algebraic curve has a unique nonsingular projective model, which means that all resolution methods are essentially the same because they all construct this model. In higher dimensions this is no longer true: varieties can have many different nonsingular projective models.
lists about 20 ways of proving resolution of singularities of curves.
Newton's method Resolution of singularities of curves was essentially first proved by , who showed the existence of Puiseux series for a curve from which resolution follows easily.
Riemann's method Riemann constructed a smooth Riemann surface from the function field of a complex algebraic curve, which gives a resolution of its singularities. This can be done over more general fields by using the set of discrete valuation rings of the field as a substitute for the Riemann surface.
Albanese's method
Albanese's method consists of taking a curve that spans a projective space of sufficiently large dimension (more than twice the degree of the curve) and repeatedly projecting down from singular points to projective spaces of smaller dimension. This method extends to higher-dimensional varieties, and shows that any n-dimensional variety has a projective model with singularities of multiplicity at most n!. For a curve, n = 1, and thus there are no singular points.
Normalization
gave a one step method of resolving singularities of a curve by taking the normalization of the curve. Normalization removes all singularities in codimension 1, so it works for curves but not in higher dimensions.
Valuation rings
Another one-step method of resolving singularities of a curve is to take a space of valuation rings of the function field of the curve. This space can be made into a nonsingular projective curve birational to the original curve.
Blowing up
Repeatedly blowing up the singular points of a curve will eventually resolve the singularities. The main task with this method is to find a way to measure the complexity of a singularity and to show that blowing up improves this measure. There are many ways to do this. For example, one can use the arithmetic genus of the curve.
Noether's method
Noether's method takes a plane curve and repeatedly applies quadratic transformations (determined by a singular point and two points in general position). Eventually this produces a plane curve whose only singularities are ordinary multiple points (all tangent lines have multiplicity two).
Bertini's method
Bertini's method is similar to Noether's method. It starts with a plane curve, and repeatedly applies birational transformations to the plane to improve the curve. The birational transformations are more complicated than the quadratic transformations used in Noether's method, but produce the better result that the only singularities are ordinary double points.
Resolution of singularities of surfaces
Surfaces have many different nonsingular projective models (unlike the case of curves where the nonsingular projective model is unique). However a surface still has a unique minimal resolution, that all others factor through (all others are resolutions of it). In higher dimensions there need not be a minimal resolution.
There were several attempts to prove resolution for surfaces over the complex numbers by , , , , and , but points out that none of these early attempts are complete, and all are vague (or even wrong) at some critical point of the argument. The first rigorous proof was given by , and an algebraic proof for all fields of characteristic 0 was given by . 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 .
Zariski's method
Zariski's 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). Although this will resolve the singularities of surfaces by itself, Zariski used a more roundabout method: he first proved a local uniformization theorem showing that every valuation of a surface could be resolved, then used the compactness of the Zariski–Riemann surface to show that it is possible to find a finite set of surfaces such that the center of each valuation is simple on at least one of these surfaces, and finally by studying birational maps between surfaces showed that this finite set of surfaces could be replaced by a single non-singular surface.
Jung's method
By applying strong embedded resolution for curves, reduces to a surface with only rather special singularities (abelian quotient singularities) which are then dealt with explicitly. The higher-dimensional version of this method is de Jong's method.
Albanese method
In general the analogue of Albanese's method for curves shows that for any variety one can reduce to singularities of order at most n!, where n is the dimension. For surfaces this reduces to the case of singularities of order 2, which are easy enough to do explicitly.
Abhyankar's method
proved resolution of singularities for surfaces over a field of any characteristic by proving a local uniformization theorem for valuation rings. The hardest case is valuation rings of rank 1 whose valuation group is a nondiscrete subgroup of the rational numbers. The rest of the proof follows Zariski's method.
Hironaka's method
Hironaka's method for arbitrary characteristic varieties gives a resolution method for surfaces, which involves repeatedly blowing up points or smooth curves in the singular set.
Lipman's method
showed that a surface Y (a 2-dimensional reduced Noetherian scheme) has a desingularization if and only if its normalization is finite over Y and analytically normal (the completions of its singular points are normal) and has only finitely many singular points. In particular if Y is excellent then it has a desingularization.
His method was to consider normal surfaces Z with a birational proper map to Y and show that there is a minimal one with minimal possible arithmetic genus. He then shows that all singularities of this minimal Z are pseudo rational, and shows that pseudo rational singularities can be resolved by repeatedly blowing up points.
Resolution of singularities in higher dimensions
The problem of resolution of singularities in higher dimensions is notorious for many incorrect published proofs and announcements of proofs that never appeared.
Zariski's method
For 3-folds the resolution of singularities was proved in characteristic 0 by . He first proved a theorem about local uniformization of valuation rings, valid for varieties of any dimension over any field of characteristic 0. He then showed that the Zariski–Riemann space of valuations is quasi-compact (for any variety of any dimension over any field), implying that there is a finite family of models of any projective variety such that any valuation has a smooth center over at least one of these models. The final and hardest part of the proof, which uses the fact that the variety is of dimension 3 but which works for all characteristics, is to show that given 2 models one can find a third that resolves the singularities that each of the two given models resolve.
Abhyankar's method
proved resolution of singularities for 3-folds in characteristic greater than 6. The restriction on the characteristic arises because Abhyankar shows that it is possible to resolve any singularity of a 3-fold of multiplicity less than the characteristic, and then uses Albanese's method to show that singularities can be reduced to those of multiplicity at most (dimension)! = 3! = 6. gave a simplified version of Abhyankar's proof.
proved resolution of singularities of 3-folds in all characteristics, by proving local uniformization in dimension at most 3, and then checking that Zariski's proof that this implies resolution for 3-folds still works in the positive characteristic case.
Hironaka's method
Resolution of singularities in characteristic 0 in all dimensions was first proved by . 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 , , , , , , . 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 and for a historical discussion see .
De Jong's method
found a different approach to resolution of singularities, generalizing Jung's method for surfaces, which was used by and by 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 is strong enough to act as a substitute for resolution for many purposes.