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 $ X $ over a ground field $ k $ is a proper birational morphism $ f : X ^ \prime \rightarrow X $ such that the variety $ X ^ \prime $ 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 $ D $ of a monoidal transformation $ X ^ \prime \rightarrow X $ is admissible (that is, $ D $ is non-singular and $ X $ is a normal flat variety along $ D $), then the numerical characteristics of the singularity of the variety (the multiplicity, the Hilbert function, etc.) are no worse than those of $ X $. The problem consists of choosing the centre of the blowing-up so that the singularities in $ X ^ \prime $ 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 $ k $ of characteristic zero has been proved. More precisely, for a reduced variety $ X _ {0} $ there exists a finite sequence of admissible monoidal transformations $ f _ {i} : X _ {i+} 1 \rightarrow X _ {i} $, $ i = 0 \dots r $, with centres $ D _ {i} \subset X _ {i} $, such that $ D _ {i} $ is contained in the set of singular points of $ X _ {i} $ and $ X _ {r} $ 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 $ \leq 3 $.

The problem of resolution of singularities is closely connected with the problem of imbedded singularities, formulated as follows. Let $ X $ be imbedded in a non-singular algebraic variety $ Z $. Does there exist a proper mapping $ f : Z ^ \prime \rightarrow Z $, with non-singular $ Z ^ \prime $, such that a) $ f $ induces an isomorphism from $ Z ^ \prime \setminus f ^ { - 1 } ( X) $ onto $ Z \setminus X $; and b) $ f ^ { - 1 } ( X) $ is a divisor with normal crossings? (A divisor on a non-singular variety has normal crossings if it is given locally by an equation $ t _ {1} \dots t _ {k} = 0 $, where $ t _ {1} \dots t _ {k} $ are part of a regular system of parameters on $ Z $.)

The problem of imbedded singularities is a particular case of the problem of trivialization of a sheaf of ideals. Let $ Z $ be a non-singular variety, let $ I $ be a coherent sheaf of ideals on $ Z $ and let $ D \subset Z $ be a non-singular closed subvariety. The weak pre-image of the ideal $ I $ under a blowing-up $ f : Z ^ \prime \rightarrow Z $ with centre in $ D $ is the sheaf of ideals

$$ f ^ { * } ( I) \otimes _ { {\mathcal O} _ {Z} } {\mathcal O} _ {Z ^ \prime } ( m D ^ \prime ) $$

on $ Z ^ \prime $, where $ D ^ \prime = f ^ { - 1 } ( D) $ and $ m $ is the multiplicity of the ideal $ I $ at a regular point of $ D $. Trivialization of a sheaf of ideals consists of finding a sequence of blowing-ups with non-singular centres for which the weak pre-image $ I $ becomes the structure sheaf. Let $ Z _ {0} $ be a non-singular variety over a field of characteristic zero, let $ I _ {0} $ be a coherent sheaf of ideals over $ Z _ {0} $ and, in addition, let there be given a certain divisor $ E _ {0} $ on $ Z _ {0} $ with normal crossings. Then there exists a sequence of blowing-ups $ f _ {i} : Z _ {i+} 1 \rightarrow Z _ {i} $, $ i = 0 \dots r - 1 $, with non-singular centres $ D _ {i} \subset Z _ {i} $, with the following properties: If $ I _ {i+} 1 $ is defined as the weak pre-image of $ I _ {i} $ under the blowing-up $ f _ {i} $ and $ E _ {i+} 1 $ is defined to be $ f _ {i} ^ { - 1 } ( E _ {i} ) \cup f _ {i} ^ { - 1 } ( D) $, then $ I _ {r} = {\mathcal O} _ {Z _ {r} } $, and $ E _ {r} $ has only normal crossings (Hironaka's theorem). In addition, one may assume that $ D _ {i} $ lies in the set of points of maximal multiplicity of $ I _ {i} $ and has normal crossings with $ E _ {i} $. For positive characteristic an analogous result is known only when $ \mathop{\rm dim} Z \leq 3 $.

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

$$ X _ {r} \rightarrow X _ {r-} 1 \rightarrow \dots \rightarrow X _ {0} = X $$

such that the induced transformation $ X _ {r} \rightarrow Y $ 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 $ \mathop{\rm char} k = 0 $ or if $ \mathop{\rm dim} X \leq 3 $.

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