View source for Blow-up algebra

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==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|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|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 [[#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.==
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|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|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====
<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>