Namespaces
Variants
Actions

Difference between revisions of "Blow-up algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
b1106401.png
 +
$#A+1 = 100 n = 0
 +
$#C+1 = 100 : ~/encyclopedia/old_files/data/B110/B.1100640 Blow\AAhup algebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
==Geometric description.==
 
==Geometric description.==
Associate to the punctured affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106401.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106402.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106403.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106404.png" />, the submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106405.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106406.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106407.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106408.png" /> varies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b1106409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064010.png" /> denotes the equivalence class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064011.png" /> in the projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064012.png" />-dimensional space. The closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064014.png" /> is smooth and is called the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064015.png" /> with centre the origin. In the real case and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064016.png" /> it is equal to the [[Möbius strip|Möbius strip]]. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064017.png" /> induced by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064018.png" /> is an isomorphism over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064019.png" />; its fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064020.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064021.png" />, the exceptional divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064022.png" />.
+
Associate to the punctured affine $  n $-
 +
space $  X _ {0} = \mathbf A  ^ {n} \setminus  \{ 0 \} $
 +
over $  \mathbf R $
 +
or $  \mathbf C $,  
 +
the submanifold $  {\widetilde{X}  } _ {0} $
 +
of $  \mathbf A  ^ {n} \times \mathbf P ^ {n - 1 } $
 +
of points $  ( x, [ x ] ) $,  
 +
where $  x $
 +
varies in $  X _ {0} $
 +
and $  [ x ] $
 +
denotes the equivalence class of $  x $
 +
in the projective $  ( n - 1 ) $-
 +
dimensional space. The closure $  {\widetilde{X}  } $
 +
of $  {\widetilde{X}  } _ {0} $
 +
is smooth and is called the blow-up of $  X = \mathbf A  ^ {n} $
 +
with centre the origin. In the real case and for $  n = 2 $
 +
it is equal to the [[Möbius strip|Möbius strip]]. The mapping $  \pi : { {\widetilde{X}  } } \rightarrow X $
 +
induced by the projection $  \mathbf A  ^ {n} \times \mathbf P ^ {n - 1 } \rightarrow \mathbf A  ^ {n} $
 +
is an isomorphism over $  X _ {0} $;  
 +
its fibre over 0 $
 +
is $  \mathbf P ^ {n - 1 } $,  
 +
the exceptional divisor of $  \pi $.
  
The strict transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064023.png" /> of a subvariety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064025.png" /> is the closure of the inverse image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064027.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064028.png" /> is the cuspidal curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064030.png" /> parametrized by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064032.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064033.png" /> and hence is smooth. This forms the simplest example of [[Resolution of singularities|resolution of singularities]] by a blow-up.
+
The strict transform $  Y  ^  \prime  $
 +
of a subvariety $  Y $
 +
of $  X $
 +
is the closure of the inverse image $  \pi ^ {-1 } ( Y \setminus  \{ 0 \} ) $
 +
in $  {\widetilde{X}  } $.  
 +
For instance, if $  Y $
 +
is the cuspidal curve $  x  ^ {3} = y  ^ {2} $
 +
in $  \mathbf A  ^ {2} $
 +
parametrized by $  ( t  ^ {2} ,t  ^ {3} ) $,  
 +
then $  Y  ^  \prime  $
 +
is given by $  ( t  ^ {2} ,t  ^ {3} ,t ) $
 +
and hence is smooth. This forms the simplest example of [[Resolution of singularities|resolution of singularities]] by a blow-up.
  
Higher-dimensional smooth centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064035.png" /> are blown up by decomposing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064036.png" /> locally along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064037.png" /> into a Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064038.png" /> of submanifolds, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064039.png" /> is transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064040.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064041.png" /> a point. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064042.png" /> is given locally as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064044.png" /> denotes the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064046.png" />.
+
Higher-dimensional smooth centres $  C $
 +
in $  \mathbf A  ^ {n} $
 +
are blown up by decomposing $  \mathbf A  ^ {n} $
 +
locally along $  C $
 +
into a Cartesian product $  X _ {1} \times X _ {2} $
 +
of submanifolds, where $  X _ {1} $
 +
is transversal to $  C $
 +
with $  X _ {1} \cap C = \{ p \} $
 +
a point. Then $  {\widetilde{X}  } $
 +
is given locally as $  { {X _ {1} } tilde } \times X _ {2} $,  
 +
where $  { {X _ {1} } tilde } $
 +
denotes the blow-up of $  X _ {1} $
 +
in $  p $.
  
 
==Algebraic description.==
 
==Algebraic description.==
See also [[#References|[a1]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064047.png" /> be a [[Noetherian ring|Noetherian ring]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064048.png" /> be an [[Ideal|ideal]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064049.png" />. Define the blow-up algebra (or Rees algebra) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064050.png" /> as the graded ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064051.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064052.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064053.png" />th power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064055.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064056.png" /> is the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064057.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064058.png" /> and coincides with the above construction when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064059.png" /> is the polynomial ring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064060.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064061.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064062.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064063.png" /> denotes the [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] given by all homogeneous prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064064.png" /> not containing the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064065.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064066.png" /> is the [[Affine variety|affine variety]] or scheme of all prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064067.png" />.
+
See also [[#References|[a1]]]. Let $  A $
 +
be a [[Noetherian ring|Noetherian ring]] and let $  I $
 +
be an [[Ideal|ideal]] of $  A $.  
 +
Define the blow-up algebra (or Rees algebra) of $  I $
 +
as the graded ring $  S = \oplus _ {k \geq  0 }  I  ^ {k} $(
 +
where $  I  ^ {k} $
 +
denotes the $  k $
 +
th power of $  I $,  
 +
$  I  ^ {0} = A $).  
 +
Then $  Bl _ {I} ( A ) = { \mathop{\rm Proj} } S $
 +
is the blow-up of $  { \mathop{\rm Spec} } A $
 +
with centre $  I $
 +
and coincides with the above construction when $  A $
 +
is the polynomial ring in $  n $
 +
variables over $  \mathbf R $
 +
or $  \mathbf C $.  
 +
Here, $  { \mathop{\rm Proj} } S $
 +
denotes the [[Algebraic variety|algebraic variety]] or [[Scheme|scheme]] given by all homogeneous prime ideals of $  S $
 +
not containing the ideal $  S _ {+} = \oplus _ {k > 0 }  I  ^ {k} $,  
 +
and $  { \mathop{\rm Spec} } A $
 +
is the [[Affine variety|affine variety]] or scheme of all prime ideals of $  A $.
  
 
==Local description.==
 
==Local description.==
Any generator system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064069.png" /> gives rise to a covering
+
Any generator system $  x _ {1} \dots x _ {k} $
 +
of $  I $
 +
gives rise to a covering
  
<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/b/b110/b110640/b11064070.png" /></td> </tr></table>
+
$$
 +
Bl _ {I} ( A ) = \cup _ {j = 1 } ^ { k }  { \mathop{\rm Spec} } A [ {I / {x _ {j} } } ] =
 +
$$
  
<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/b/b110/b110640/b11064071.png" /></td> </tr></table>
+
$$
 +
=  
 +
\cup _ {j = 1 } ^ { k }  { \mathop{\rm Spec} } A [ { {x _ {i} } / {x _ {j} } } ,  1 \leq  i \leq  k ]
 +
$$
  
by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064072.png" /> affine charts, the quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064073.png" /> being considered as elements of the localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064074.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064075.png" /> (cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]). In the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064076.png" />th chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064077.png" />, the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064078.png" /> is induced by the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064079.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064080.png" /> an ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064081.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064082.png" />, the strict transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064083.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064084.png" />. The exceptional divisor has the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064085.png" />. If the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064086.png" /> given by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064087.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064088.png" /> is smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064089.png" /> is generated by part of a regular parameter system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064091.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064092.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064094.png" />, and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064095.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064096.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064097.png" />.
+
by $  k $
 +
affine charts, the quotients $  { {x _ {i} } / {x _ {j} } } $
 +
being considered as elements of the localization of $  A $
 +
at $  x _ {j} $(
 +
cf. [[Localization in a commutative algebra|Localization in a commutative algebra]]). In the $  j $
 +
th chart $  {\widetilde{X}  } _ {j} $,  
 +
the morphism $  \pi : { {\widetilde{X}  } _ {j} } \rightarrow X $
 +
is induced by the inclusion $  A \subset  A [ {I / {x _ {j} } } ] $.  
 +
For $  J $
 +
an ideal of $  A $
 +
contained in $  I $,  
 +
the strict transform of $  J $
 +
is $  J  ^  \prime  = \cup _ {n \geq 0 }  x _ {j} ^ {- n } ( J \cap I  ^ {n} ) {\widetilde{A}  } _ {j} $.  
 +
The exceptional divisor has the equation $  x _ {j} = 0 $.  
 +
If the centre $  C $
 +
given by the ideal $  I $
 +
of $  A $
 +
is smooth, $  I $
 +
is generated by part of a regular parameter system of $  A $
 +
and $  \pi : { {\widetilde{X}  } _ {j} } \rightarrow X $
 +
is given by $  x _ {i} \rightarrow x _ {i} x _ {j} $
 +
for $  i \leq  k $,  
 +
$  i \neq j $,  
 +
and by $  x _ {i} \rightarrow x _ {i} $
 +
for $  i > k $
 +
or $  i = j $.
  
 
==Properties.==
 
==Properties.==
Different centres may induce the same blow-up. A composite of blow-ups is again a blow-up. Blowing up commutes with [[Base change|base change]]; the strict transform of a variety equals its blow-up in the given centre. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064098.png" /> is birational, proper and surjective (cf. [[Birational morphism|Birational morphism]]; [[Proper morphism|Proper morphism]]; [[Surjection|Surjection]]). Any birational projective morphism of quasi-projective varieties (cf. [[Quasi-projective scheme|Quasi-projective scheme]]) is the blowing up of a suitable centre. The singularities of varieties over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b11064099.png" /> can be resolved by a finite sequence of blow-ups of smooth centres [[#References|[a2]]]. In positive characteristic, this has only been proven for dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110640/b110640100.png" /> [[#References|[a3]]]. See [[#References|[a4]]] for a survey on resolution of singularities, and [[#References|[a5]]] for an account on the role of blow-up algebras in commutative algebra.
+
Different centres may induce the same blow-up. A composite of blow-ups is again a blow-up. Blowing up commutes with [[Base change|base change]]; the strict transform of a variety equals its blow-up in the given centre. The morphism $  \pi $
 +
is birational, proper and surjective (cf. [[Birational morphism|Birational morphism]]; [[Proper morphism|Proper morphism]]; [[Surjection|Surjection]]). Any birational projective morphism of quasi-projective varieties (cf. [[Quasi-projective scheme|Quasi-projective scheme]]) is the blowing up of a suitable centre. The singularities of varieties over a field of characteristic 0 $
 +
can be resolved by a finite sequence of blow-ups of smooth centres [[#References|[a2]]]. In positive characteristic, this has only been proven for dimension $  \leq  3 $[[#References|[a3]]]. See [[#References|[a4]]] for a survey on resolution of singularities, and [[#References|[a5]]] for an account on the role of blow-up algebras in commutative algebra.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" ''Ann. of Math.'' , '''79''' (1964) pp. 109–326 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Abhyankar, "Resolution of singularities of embedded algebraic surfaces" , Acad. Press (1966) {{MR|0217069}} {{ZBL|0147.20504}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Lipman, "Introduction to resolution of singularities" , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1975) pp. 187–230 {{MR|0389901}} {{ZBL|0306.14007}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Vasconcelos, "Arithmetic of blowup algebras" , ''Lecture Notes Ser.'' , '''195''' , London Math. Soc. (1994) {{MR|1275840}} {{ZBL|0813.13008}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" ''Ann. of Math.'' , '''79''' (1964) pp. 109–326 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Abhyankar, "Resolution of singularities of embedded algebraic surfaces" , Acad. Press (1966) {{MR|0217069}} {{ZBL|0147.20504}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> J. Lipman, "Introduction to resolution of singularities" , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc. (1975) pp. 187–230 {{MR|0389901}} {{ZBL|0306.14007}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Vasconcelos, "Arithmetic of blowup algebras" , ''Lecture Notes Ser.'' , '''195''' , London Math. Soc. (1994) {{MR|1275840}} {{ZBL|0813.13008}} </TD></TR></table>

Latest revision as of 10:59, 29 May 2020


Geometric description.

Associate to the punctured affine $ n $- space $ X _ {0} = \mathbf A ^ {n} \setminus \{ 0 \} $ over $ \mathbf R $ or $ \mathbf C $, the submanifold $ {\widetilde{X} } _ {0} $ of $ \mathbf A ^ {n} \times \mathbf P ^ {n - 1 } $ of points $ ( x, [ x ] ) $, where $ x $ varies in $ X _ {0} $ and $ [ x ] $ denotes the equivalence class of $ x $ in the projective $ ( n - 1 ) $- dimensional space. The closure $ {\widetilde{X} } $ of $ {\widetilde{X} } _ {0} $ is smooth and is called the blow-up of $ X = \mathbf A ^ {n} $ with centre the origin. In the real case and for $ n = 2 $ it is equal to the Möbius strip. The mapping $ \pi : { {\widetilde{X} } } \rightarrow X $ induced by the projection $ \mathbf A ^ {n} \times \mathbf P ^ {n - 1 } \rightarrow \mathbf A ^ {n} $ is an isomorphism over $ X _ {0} $; its fibre over $ 0 $ is $ \mathbf P ^ {n - 1 } $, the exceptional divisor of $ \pi $.

The strict transform $ Y ^ \prime $ of a subvariety $ Y $ of $ X $ is the closure of the inverse image $ \pi ^ {-1 } ( Y \setminus \{ 0 \} ) $ in $ {\widetilde{X} } $. For instance, if $ Y $ is the cuspidal curve $ x ^ {3} = y ^ {2} $ in $ \mathbf A ^ {2} $ parametrized by $ ( t ^ {2} ,t ^ {3} ) $, then $ Y ^ \prime $ is given by $ ( t ^ {2} ,t ^ {3} ,t ) $ and hence is smooth. This forms the simplest example of resolution of singularities by a blow-up.

Higher-dimensional smooth centres $ C $ in $ \mathbf A ^ {n} $ are blown up by decomposing $ \mathbf A ^ {n} $ locally along $ C $ into a Cartesian product $ X _ {1} \times X _ {2} $ of submanifolds, where $ X _ {1} $ is transversal to $ C $ with $ X _ {1} \cap C = \{ p \} $ a point. Then $ {\widetilde{X} } $ is given locally as $ { {X _ {1} } tilde } \times X _ {2} $, where $ { {X _ {1} } tilde } $ denotes the blow-up of $ X _ {1} $ in $ p $.

Algebraic description.

See also [a1]. Let $ A $ be a Noetherian ring and let $ I $ be an ideal of $ A $. Define the blow-up algebra (or Rees algebra) of $ I $ as the graded ring $ S = \oplus _ {k \geq 0 } I ^ {k} $( where $ I ^ {k} $ denotes the $ k $ th power of $ I $, $ I ^ {0} = A $). Then $ Bl _ {I} ( A ) = { \mathop{\rm Proj} } S $ is the blow-up of $ { \mathop{\rm Spec} } A $ with centre $ I $ and coincides with the above construction when $ A $ is the polynomial ring in $ n $ variables over $ \mathbf R $ or $ \mathbf C $. Here, $ { \mathop{\rm Proj} } S $ denotes the algebraic variety or scheme given by all homogeneous prime ideals of $ S $ not containing the ideal $ S _ {+} = \oplus _ {k > 0 } I ^ {k} $, and $ { \mathop{\rm Spec} } A $ is the affine variety or scheme of all prime ideals of $ A $.

Local description.

Any generator system $ x _ {1} \dots x _ {k} $ of $ I $ gives rise to a covering

$$ Bl _ {I} ( A ) = \cup _ {j = 1 } ^ { k } { \mathop{\rm Spec} } A [ {I / {x _ {j} } } ] = $$

$$ = \cup _ {j = 1 } ^ { k } { \mathop{\rm Spec} } A [ { {x _ {i} } / {x _ {j} } } , 1 \leq i \leq k ] $$

by $ k $ affine charts, the quotients $ { {x _ {i} } / {x _ {j} } } $ being considered as elements of the localization of $ A $ at $ x _ {j} $( cf. Localization in a commutative algebra). In the $ j $ th chart $ {\widetilde{X} } _ {j} $, the morphism $ \pi : { {\widetilde{X} } _ {j} } \rightarrow X $ is induced by the inclusion $ A \subset A [ {I / {x _ {j} } } ] $. For $ J $ an ideal of $ A $ contained in $ I $, the strict transform of $ J $ is $ J ^ \prime = \cup _ {n \geq 0 } x _ {j} ^ {- n } ( J \cap I ^ {n} ) {\widetilde{A} } _ {j} $. The exceptional divisor has the equation $ x _ {j} = 0 $. If the centre $ C $ given by the ideal $ I $ of $ A $ is smooth, $ I $ is generated by part of a regular parameter system of $ A $ and $ \pi : { {\widetilde{X} } _ {j} } \rightarrow X $ is given by $ x _ {i} \rightarrow x _ {i} x _ {j} $ for $ i \leq k $, $ i \neq j $, and by $ x _ {i} \rightarrow x _ {i} $ for $ i > k $ or $ i = j $.

Properties.

Different centres may induce the same blow-up. A composite of blow-ups is again a blow-up. Blowing up commutes with base change; the strict transform of a variety equals its blow-up in the given centre. The morphism $ \pi $ is birational, proper and surjective (cf. Birational morphism; Proper morphism; Surjection). Any birational projective morphism of quasi-projective varieties (cf. Quasi-projective scheme) is the blowing up of a suitable centre. The singularities of varieties over a field of characteristic $ 0 $ can be resolved by a finite sequence of blow-ups of smooth centres [a2]. In positive characteristic, this has only been proven for dimension $ \leq 3 $[a3]. See [a4] for a survey on resolution of singularities, and [a5] for an account on the role of blow-up algebras in commutative algebra.

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[a2] H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" Ann. of Math. , 79 (1964) pp. 109–326 MR0199184 Zbl 0122.38603
[a3] S. Abhyankar, "Resolution of singularities of embedded algebraic surfaces" , Acad. Press (1966) MR0217069 Zbl 0147.20504
[a4] J. Lipman, "Introduction to resolution of singularities" , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 187–230 MR0389901 Zbl 0306.14007
[a5] W. Vasconcelos, "Arithmetic of blowup algebras" , Lecture Notes Ser. , 195 , London Math. Soc. (1994) MR1275840 Zbl 0813.13008
How to Cite This Entry:
Blow-up algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blow-up_algebra&oldid=23768
This article was adapted from an original article by H. Hauser (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article