# 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 $\mathop{\rm dim} X \leq 3$.

How to Cite This Entry:
Resolution of singularities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolution_of_singularities&oldid=48528
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article