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''blowing up, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647502.png" />-process''
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A special kind of [[Birational morphism|birational morphism]] of an algebraic variety or bimeromorphic morphism of an analytic space. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647503.png" /> be an algebraic variety (or an arbitrary scheme), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647504.png" /> be a closed subvariety given by a sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647505.png" />. The monoidal transformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647506.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647507.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647508.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m0647509.png" /> — the projective spectrum of the graded sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475010.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475012.png" /> is the structure morphism of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475013.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475014.png" />, then the sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475016.png" /> (defining the exceptional subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475017.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475018.png" />) is invertible. This means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475019.png" /> is a [[Divisor|divisor]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475020.png" />; in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475021.png" /> induces an isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475023.png" />. A monoidal transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475024.png" /> of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475025.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475026.png" /> is characterized by the following universal property [[#References|[1]]]: The sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475027.png" /> is invertible and for any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475028.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475029.png" /> is invertible there is a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475031.png" />.
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A monoidal transformation of an algebraic or analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475032.png" /> with as centre a closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475033.png" /> can be defined and characterized in the same way.
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''blowing up,  $  \sigma $-
 +
process''
  
An important class of monoidal transformations are the admissible monoidal transformations, which are distinguished by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475034.png" /> is non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475035.png" /> is a normally flat scheme along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475036.png" />. The latter means that all sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475037.png" /> are flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475038.png" />-modules. The importance of admissible monoidal transformations is explained by the fact that they do not worsen the singularities of the variety. In addition, it has been proved (see [[#References|[1]]]) that a suitable sequence of admissible monoidal transformations improves singularities, which allows one to prove the theorem on the resolution of singularities for an algebraic variety over a field of characteristic zero.
+
A special kind of [[Birational morphism|birational morphism]] of an algebraic variety or bimeromorphic morphism of an analytic space. For example, let  $  X $
 +
be an algebraic variety (or an arbitrary scheme), and let  $  D \subset  X $
 +
be a closed subvariety given by a sheaf of ideals  $  J $.
 +
The monoidal transformation of  $  X $
 +
with centre  $  D $
 +
is the $  X $-
 +
scheme  $  X  ^ {1} = \mathop{\rm Proj} ( \oplus _ {n \geq  0 }  J  ^ {n)} $—
 +
the projective spectrum of the graded sheaf of  $  {\mathcal O} _ {X} $-
 +
algebras  $  \oplus _ {n \geq  0 }  J  ^ {n} $.
 +
If  $  f : X  ^ {1} \rightarrow X $
 +
is the structure morphism of the  $  X $-
 +
scheme $  X  ^ {1} $,
 +
then the sheaf of ideals  $  f ^ { * } ( J) = J \cdot {\mathcal O} _ {X  ^ {1}  } $
 +
on  $  X  ^ {1} $(
 +
defining the exceptional subscheme  $  f ^ { - 1 } ( D ) $
 +
on  $  X  ^ {1} $)
 +
is invertible. This means that $  f ^ { - 1 } ( D ) $
 +
is a [[Divisor|divisor]] on  $  X  ^ {1} $;
 +
in addition,  $  f $
 +
induces an isomorphism between  $  X  ^ {1} \setminus  f ^ { - 1 } ( D ) $
 +
and  $  X \setminus  D $.  
 +
A monoidal transformation  $  f : X  ^ {1} \rightarrow X $
 +
of a scheme  $  X $
 +
with centre  $  D $
 +
is characterized by the following universal property [[#References|[1]]]: The sheaf of ideals  $  f ^ { * } ( J ) $
 +
is invertible and for any morphism  $  g :  X _ {1} \rightarrow X $
 +
for which $  g  ^ {*} ( J ) $
 +
is invertible there is a unique morphism  $  h :  X _ {1} \rightarrow X  ^ {1} $
 +
such that  $  g = f \circ h $.
  
Admissible monoidal transformations of non-singular varieties are particularly simple to construct. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475039.png" /> is a monoidal transformation with a non-singular centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475041.png" /> is again non-singular and the exceptional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475042.png" /> is canonically isomorphic to the projectivization of the conormal sheaf to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475043.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475044.png" />. In the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475045.png" /> consists of one point, the monoidal transformation consists of blowing up this point into the whole projective space of tangent directions. For the behaviour of various invariants of non-singular varieties (such as Chow rings, cohomology spaces, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475046.png" />-functor, and Chern classes) under admissible monoidal transformations see [[#References|[2]]]–[[#References|[5]]].
+
A monoidal transformation of an algebraic or analytic space  $  X $
 +
with as centre a closed subspace  $  D \subset  X $
 +
can be defined and characterized in the same way.
  
====References====
+
An important class of monoidal transformations are the admissible monoidal transformations, which are distinguished by the condition that  $  D $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Hironaka, "Resolution of an algebraic variety over a field of characteristic zero I, II" ''Ann. of Math. (2)'' , '''79''' (1964) pp. 109–326 {{MR|199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , ''Théorie des intersections et théorème de Riemann–Roch (SGA 6)'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475047.png" />-adique et fonctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475048.png" />. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer (1977) {{MR|491704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I. Porteous, "Blowing up Chern classes" ''Proc. Cambridge Philos. Soc.'' , '''56''' : 2 (1960) pp. 118–124 {{MR|0121813}} {{ZBL|0166.16701}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064750/m06475049.png" />-functor in algebraic geometry" ''Russian Math. Surveys'' , '''24''' (1969) pp. 1–89 ''Uspekhi Mat. Nauk'' , '''24''' : 5 (1969) pp. 3–86 {{MR|265355}} {{ZBL|}} </TD></TR></table>
+
is non-singular and  $  X $
 +
is a normally flat scheme along  $  D $.  
 +
The latter means that all sheaves  $  J  ^ {n} / J  ^ {n+} 1 $
 +
are flat  $  ( {\mathcal O} _ {X} / J ) $-
 +
modules. The importance of admissible monoidal transformations is explained by the fact that they do not worsen the singularities of the variety. In addition, it has been proved (see [[#References|[1]]]) that a suitable sequence of admissible monoidal transformations improves singularities, which allows one to prove the theorem on the resolution of singularities for an algebraic variety over a field of characteristic zero.
  
 +
Admissible monoidal transformations of non-singular varieties are particularly simple to construct. If  $  f :  X _ {1} \rightarrow X $
 +
is a monoidal transformation with a non-singular centre  $  D \subset  X $,
 +
then  $  X _ {1} $
 +
is again non-singular and the exceptional subspace  $  f ^ { - 1 } ( D ) $
 +
is canonically isomorphic to the projectivization of the conormal sheaf to  $  D $
 +
in  $  X $.
 +
In the special case when  $  D $
 +
consists of one point, the monoidal transformation consists of blowing up this point into the whole projective space of tangent directions. For the behaviour of various invariants of non-singular varieties (such as Chow rings, cohomology spaces, the  $  K $-
 +
functor, and Chern classes) under admissible monoidal transformations see [[#References|[2]]]–[[#References|[5]]].
  
  
 
====Comments====
 
====Comments====
 +
 
The word "s-process" appeared for the first time in [[#References|[a1]]].
 
The word "s-process" appeared for the first time in [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Hopf, "Schlichte Abbildungen und lokale Modifikationen 4-dimensionaler komplexer Mannigfaltigkeiten" ''Comm. Math. Helv.'' , '''29''' (1954) pp. 132–156 {{MR|0068008}} {{ZBL|0064.41703}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> H. Hironaka, "Resolution of an algebraic variety over a field of characteristic zero I, II" ''Ann. of Math. (2)'' , '''79''' (1964) pp. 109–326 {{MR|199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , ''Théorie des intersections et théorème de Riemann–Roch (SGA 6)'' , ''Lect. notes in math.'' , '''225''' , Springer (1971)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , ''Lect. notes in math.'' , '''589''' , Springer (1977) {{MR|491704}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I. Porteous, "Blowing up Chern classes" ''Proc. Cambridge Philos. Soc.'' , '''56''' : 2 (1960) pp. 118–124 {{MR|0121813}} {{ZBL|0166.16701}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> Yu.I. Manin, "Lectures on the $K$-functor in algebraic geometry" ''Russian Math. Surveys'' , '''24''' (1969) pp. 1–89 ''Uspekhi Mat. Nauk'' , '''24''' : 5 (1969) pp. 3–86 {{MR|265355}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Hopf, "Schlichte Abbildungen und lokale Modifikationen 4-dimensionaler komplexer Mannigfaltigkeiten" ''Comm. Math. Helv.'' , '''29''' (1954) pp. 132–156 {{MR|0068008}} {{ZBL|0064.41703}} </TD></TR>
 +
</table>
  
 
[[Category:Algebraic geometry]]
 
[[Category:Algebraic geometry]]

Latest revision as of 11:53, 8 April 2023


blowing up, $ \sigma $- process

A special kind of birational morphism of an algebraic variety or bimeromorphic morphism of an analytic space. For example, let $ X $ be an algebraic variety (or an arbitrary scheme), and let $ D \subset X $ be a closed subvariety given by a sheaf of ideals $ J $. The monoidal transformation of $ X $ with centre $ D $ is the $ X $- scheme $ X ^ {1} = \mathop{\rm Proj} ( \oplus _ {n \geq 0 } J ^ {n)} $— the projective spectrum of the graded sheaf of $ {\mathcal O} _ {X} $- algebras $ \oplus _ {n \geq 0 } J ^ {n} $. If $ f : X ^ {1} \rightarrow X $ is the structure morphism of the $ X $- scheme $ X ^ {1} $, then the sheaf of ideals $ f ^ { * } ( J) = J \cdot {\mathcal O} _ {X ^ {1} } $ on $ X ^ {1} $( defining the exceptional subscheme $ f ^ { - 1 } ( D ) $ on $ X ^ {1} $) is invertible. This means that $ f ^ { - 1 } ( D ) $ is a divisor on $ X ^ {1} $; in addition, $ f $ induces an isomorphism between $ X ^ {1} \setminus f ^ { - 1 } ( D ) $ and $ X \setminus D $. A monoidal transformation $ f : X ^ {1} \rightarrow X $ of a scheme $ X $ with centre $ D $ is characterized by the following universal property [1]: The sheaf of ideals $ f ^ { * } ( J ) $ is invertible and for any morphism $ g : X _ {1} \rightarrow X $ for which $ g ^ {*} ( J ) $ is invertible there is a unique morphism $ h : X _ {1} \rightarrow X ^ {1} $ such that $ g = f \circ h $.

A monoidal transformation of an algebraic or analytic space $ X $ with as centre a closed subspace $ D \subset X $ can be defined and characterized in the same way.

An important class of monoidal transformations are the admissible monoidal transformations, which are distinguished by the condition that $ D $ is non-singular and $ X $ is a normally flat scheme along $ D $. The latter means that all sheaves $ J ^ {n} / J ^ {n+} 1 $ are flat $ ( {\mathcal O} _ {X} / J ) $- modules. The importance of admissible monoidal transformations is explained by the fact that they do not worsen the singularities of the variety. In addition, it has been proved (see [1]) that a suitable sequence of admissible monoidal transformations improves singularities, which allows one to prove the theorem on the resolution of singularities for an algebraic variety over a field of characteristic zero.

Admissible monoidal transformations of non-singular varieties are particularly simple to construct. If $ f : X _ {1} \rightarrow X $ is a monoidal transformation with a non-singular centre $ D \subset X $, then $ X _ {1} $ is again non-singular and the exceptional subspace $ f ^ { - 1 } ( D ) $ is canonically isomorphic to the projectivization of the conormal sheaf to $ D $ in $ X $. In the special case when $ D $ consists of one point, the monoidal transformation consists of blowing up this point into the whole projective space of tangent directions. For the behaviour of various invariants of non-singular varieties (such as Chow rings, cohomology spaces, the $ K $- functor, and Chern classes) under admissible monoidal transformations see [2][5].


Comments

The word "s-process" appeared for the first time in [a1].

References

[1] H. Hironaka, "Resolution of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. (2) , 79 (1964) pp. 109–326 MR199184 Zbl 0122.38603
[2] P. Berthelot (ed.) A. Grothendieck (ed.) L. Illusie (ed.) et al. (ed.) , Théorie des intersections et théorème de Riemann–Roch (SGA 6) , Lect. notes in math. , 225 , Springer (1971)
[3] A. Grothendieck, I. Bucur, C. Honzel, L. Illusie, J.-P. Jouanolou, J.-P. Serre, "Cohomologie $\ell$-adique et fonctions $L$. SGA 5" , Lect. notes in math. , 589 , Springer (1977) MR491704
[4] I. Porteous, "Blowing up Chern classes" Proc. Cambridge Philos. Soc. , 56 : 2 (1960) pp. 118–124 MR0121813 Zbl 0166.16701
[5] Yu.I. Manin, "Lectures on the $K$-functor in algebraic geometry" Russian Math. Surveys , 24 (1969) pp. 1–89 Uspekhi Mat. Nauk , 24 : 5 (1969) pp. 3–86 MR265355
[a1] H. Hopf, "Schlichte Abbildungen und lokale Modifikationen 4-dimensionaler komplexer Mannigfaltigkeiten" Comm. Math. Helv. , 29 (1954) pp. 132–156 MR0068008 Zbl 0064.41703
How to Cite This Entry:
Monoidal transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Monoidal_transformation&oldid=34206
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article