Namespaces
Variants
Actions

Difference between revisions of "Modification"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
m0644001.png
 +
$#A+1 = 7 n = 0
 +
$#C+1 = 7 : ~/encyclopedia/old_files/data/M064/M.0604400 Modification
 +
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}}
 +
 
''of an analytic space''
 
''of an analytic space''
  
An analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064400/m0644001.png" /> of analytic spaces such that for certain analytic sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064400/m0644002.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064400/m0644003.png" /> of smaller dimensions, the conditions
+
An analytic mapping $  f : X \rightarrow Y $
 +
of analytic spaces such that for certain analytic sets $  S \subset  X $
 +
and $  T \subset  Y $
 +
of smaller dimensions, the conditions
  
<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/m/m064/m064400/m0644004.png" /></td> </tr></table>
+
$$
 +
f : X \setminus  S  \rightarrow  Y \setminus  T \ \
 +
\textrm{ is  an  isomorphism  }
 +
$$
  
 
and
 
and
  
<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/m/m064/m064400/m0644005.png" /></td> </tr></table>
+
$$
 +
f ( S)  = T
 +
$$
  
hold. A modification is also called a contraction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064400/m0644006.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064400/m0644007.png" />. An example of a modification is a [[Monoidal transformation|monoidal transformation]].
+
hold. A modification is also called a contraction of $  S $
 +
onto $  T $.  
 +
An example of a modification is a [[Monoidal transformation|monoidal transformation]].
  
 
See also [[Exceptional analytic set|Exceptional analytic set]]; [[Exceptional subvariety|Exceptional subvariety]].
 
See also [[Exceptional analytic set|Exceptional analytic set]]; [[Exceptional subvariety|Exceptional subvariety]].
Line 15: Line 37:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Behnke, K. Stein, "Modifikation komplexer Mannigfaltigkeiten und Riemannschen Gebiete" ''Math. Ann.'' , '''124''' : 1 (1951) pp. 1–16</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Behnke, K. Stein, "Modifikation komplexer Mannigfaltigkeiten und Riemannschen Gebiete" ''Math. Ann.'' , '''124''' : 1 (1951) pp. 1–16</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====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></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 08:01, 6 June 2020


of an analytic space

An analytic mapping $ f : X \rightarrow Y $ of analytic spaces such that for certain analytic sets $ S \subset X $ and $ T \subset Y $ of smaller dimensions, the conditions

$$ f : X \setminus S \rightarrow Y \setminus T \ \ \textrm{ is an isomorphism } $$

and

$$ f ( S) = T $$

hold. A modification is also called a contraction of $ S $ onto $ T $. An example of a modification is a monoidal transformation.

See also Exceptional analytic set; Exceptional subvariety.

References

[1] H. Behnke, K. Stein, "Modifikation komplexer Mannigfaltigkeiten und Riemannschen Gebiete" Math. Ann. , 124 : 1 (1951) pp. 1–16

Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
How to Cite This Entry:
Modification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Modification&oldid=23900
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article