Difference between revisions of "Modification"
From Encyclopedia of Mathematics
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
Ulf Rehmann (talk | contribs) 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 | + | 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 | and | ||
| − | + | $$ | |
| + | f ( S) = T | ||
| + | $$ | ||
| − | hold. A modification is also called a contraction of | + | 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
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