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''desingularization''
 
''desingularization''
  
The replacement of a singular [[Algebraic variety|algebraic variety]] by a birationally isomorphic non-singular variety. More precisely, a resolution of singularities of an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815801.png" /> over a ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815802.png" /> is a proper birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815803.png" /> such that the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815804.png" /> is non-singular (smooth) (cf. [[Proper morphism|Proper morphism]]; [[Birational 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.
+
The replacement of a singular [[Algebraic variety|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|Proper morphism]]; [[Birational 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|Monoidal transformation]]). It is known that if the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815805.png" /> of a monoidal transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815806.png" /> is admissible (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815807.png" /> is non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815808.png" /> is a normal flat variety along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815809.png" />), then the numerical characteristics of the singularity of the variety (the multiplicity, the Hilbert function, etc.) are no worse than those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158010.png" />. The problem consists of choosing the centre of the blowing-up so that the singularities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158011.png" /> really are improved.
+
Usually a resolution of singularities is the result of successive application of monoidal transformations (cf. [[Monoidal transformation|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158012.png" /> of characteristic zero has been proved. More precisely, for a reduced variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158013.png" /> there exists a finite sequence of admissible monoidal transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158015.png" />, with centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158016.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158017.png" /> is contained in the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158019.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158020.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158021.png" /> be imbedded in a non-singular algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158022.png" />. Does there exist a proper mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158023.png" />, with non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158024.png" />, such that a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158025.png" /> induces an isomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158026.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158027.png" />; and b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158028.png" /> is a divisor with normal crossings? (A divisor on a non-singular variety has normal crossings if it is given locally by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158030.png" /> are part of a regular system of parameters on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158031.png" />.)
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158032.png" /> be a non-singular variety, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158033.png" /> be a [[Coherent sheaf|coherent sheaf]] of ideals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158034.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158035.png" /> be a non-singular closed subvariety. The weak pre-image of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158036.png" /> under a blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158037.png" /> with centre in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158038.png" /> is the sheaf of ideals
+
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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158039.png" /></td> </tr></table>
+
$$
 +
f ^ { * } ( I) \otimes _ { {\mathcal O} _ {Z}  } {\mathcal O} _ {Z  ^  \prime  } ( m D  ^  \prime  )
 +
$$
  
on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158042.png" /> is the multiplicity of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158043.png" /> at a regular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158044.png" />. Trivialization of a sheaf of ideals consists of finding a sequence of blowing-ups with non-singular centres for which the weak pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158045.png" /> becomes the structure sheaf. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158046.png" /> be a non-singular variety over a field of characteristic zero, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158047.png" /> be a coherent sheaf of ideals over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158048.png" /> and, in addition, let there be given a certain divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158050.png" /> with normal crossings. Then there exists a sequence of blowing-ups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158052.png" />, with non-singular centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158053.png" />, with the following properties: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158054.png" /> is defined as the weak pre-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158055.png" /> under the blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158057.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158060.png" /> has only normal crossings (Hironaka's theorem). In addition, one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158061.png" /> lies in the set of points of maximal multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158062.png" /> and has normal crossings with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158063.png" />. For positive characteristic an analogous result is known only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158064.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158065.png" /> be a rational transformation of non-singular algebraic varieties. Does there exist a sequence of blowing-ups with non-singular centres
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158066.png" /></td> </tr></table>
+
$$
 +
X _ {r}  \rightarrow  X _ {r-} 1  \rightarrow \dots \rightarrow  X _ {0= X
 +
$$
  
such that the induced transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158067.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158068.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158069.png" />.
+
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====
 
====References====

Latest revision as of 08:11, 6 June 2020


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
How to Cite This Entry:
Resolution of singularities. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolution_of_singularities&oldid=34219
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article