Resolution of singularities

desingularization

The replacement of a singular algebraic variety by a birationally isomorphic non-singular variety. More precisely, a resolution of singularities of an algebraic variety over a ground field is a proper birational morphism such that the variety is non-singular (smooth) (cf. Proper morphism; Birational morphism). A resolution of singularities of a scheme, of a complex-analytic space, etc. is defined analogously. The existence of a resolution of singularities enables one to reduce many questions to non-singular varieties, and in studying these one can use intersection theory and the apparatus of differential forms.

Usually a resolution of singularities is the result of successive application of monoidal transformations (cf. Monoidal transformation). It is known that if the centre of a monoidal transformation is admissible (that is, is non-singular and is a normal flat variety along ), then the numerical characteristics of the singularity of the variety (the multiplicity, the Hilbert function, etc.) are no worse than those of . The problem consists of choosing the centre of the blowing-up so that the singularities in really are improved.

In the case of curves the problem of resolution of singularities essentially reduces to normalization. In the two-dimensional case the situation is more complicated. The existence of a resolution of singularities for any variety over a field of characteristic zero has been proved. More precisely, for a reduced variety there exists a finite sequence of admissible monoidal transformations , , with centres , such that is contained in the set of singular points of and is a non-singular variety. An analogous result is true for complex-analytic spaces. In positive characteristic the existence of a resolution of singularities has been established (1983) for dimensions .

The problem of resolution of singularities is closely connected with the problem of imbedded singularities, formulated as follows. Let be imbedded in a non-singular algebraic variety . Does there exist a proper mapping , with non-singular , such that a) induces an isomorphism from onto ; and b) is a divisor with normal crossings? (A divisor on a non-singular variety has normal crossings if it is given locally by an equation , where are part of a regular system of parameters on .)

The problem of imbedded singularities is a particular case of the problem of trivialization of a sheaf of ideals. Let be a non-singular variety, let be a coherent sheaf of ideals on and let be a non-singular closed subvariety. The weak pre-image of the ideal under a blowing-up with centre in is the sheaf of ideals

on , where and is the multiplicity of the ideal at a regular point of . Trivialization of a sheaf of ideals consists of finding a sequence of blowing-ups with non-singular centres for which the weak pre-image becomes the structure sheaf. Let be a non-singular variety over a field of characteristic zero, let be a coherent sheaf of ideals over and, in addition, let there be given a certain divisor on with normal crossings. Then there exists a sequence of blowing-ups , , with non-singular centres , with the following properties: If is defined as the weak pre-image of under the blowing-up and is defined to be , then , and has only normal crossings (Hironaka's theorem). In addition, one may assume that lies in the set of points of maximal multiplicity of and has normal crossings with . For positive characteristic an analogous result is known only when .

Another problem of this type is the problem of eliminating points of indeterminacy of a rational transformation. Let be a rational transformation of non-singular algebraic varieties. Does there exist a sequence of blowing-ups with non-singular centres

such that the induced transformation is a morphism? This problem reduces to the problem of the existence of a trivialization of a sheaf of ideals, and the answer is affirmative if or if .

References

[1]	S.S. Abhyankar, "Resolution of singularities of embedded algebraic surfaces" , Acad. Press (1966) MR0217069 Zbl 0147.20504
[2]	J. Lipman, "Introduction to resolution of singularities" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 187–230 MR0389901 Zbl 0306.14007
[3]	H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. , 79 (1964) pp. 109–326