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
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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 |
Resolution of singularities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolution_of_singularities&oldid=34219