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Difference between revisions of "Segre imbedding"

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The imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838001.png" /> of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838002.png" /> of projective spaces into the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838003.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838004.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838006.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838007.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838008.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s0838009.png" />) are homogeneous coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380010.png" />, then the mapping is defined by the formula:
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<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/s/s083/s083800/s08380011.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380012.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380013.png" /> is well-defined and is a closed imbedding. The image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380014.png" /> of a Segre imbedding is called a Segre variety. The case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380015.png" /> has a simple geometrical meaning: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380016.png" /> is the non-singular quadric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380017.png" /> with equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380018.png" />. The images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083800/s08380020.png" /> give two families of generating lines of the quadric.
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The imbedding  $  \phi :  P  ^ {n} \times P  ^ {m} \rightarrow P  ^ {N} $
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of the product  $  P  ^ {n} \times P  ^ {m} $
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of projective spaces into the projective space  $  P  ^ {N} $,
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where $  N = nm + n + m $.
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If  $  x = ( u _ {0} : \dots : u _ {n} ) \in P  ^ {n} $,
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$  y = ( v _ {0} : \dots : v _ {m} ) \in P  ^ {m} $,
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and  $  w _ {i,j} $(
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$  i = 0 \dots n $;
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$  j = 0 \dots m $)
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are homogeneous coordinates in  $  P  ^ {N} $,
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then the mapping is defined by the formula:
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$$
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\phi ( x , y)  = ( w _ {i,j} )  \in  P  ^ {N} ,
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$$
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where  $  w _ {i,j} = u _ {i} v _ {j} $.  
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The mapping  $  \phi $
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is well-defined and is a closed imbedding. The image $  \phi ( P  ^ {n} \times P  ^ {m} ) $
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of a Segre imbedding is called a Segre variety. The case when $  n = m = 1 $
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has a simple geometrical meaning: $  \phi ( P  ^ {1} \times P  ^ {1} ) $
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is the non-singular quadric in $  P  ^ {3} $
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with equation $  w _ {11} w _ {00} = w _ {01} w _ {10} $.  
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The images $  \phi ( x \times P  ^ {1} ) $
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and $  \phi ( P  ^ {1} \times y) $
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give two families of generating lines of the quadric.
  
 
The terminology is in honour of B. Segre.
 
The terminology is in honour of B. Segre.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 08:12, 6 June 2020


The imbedding $ \phi : P ^ {n} \times P ^ {m} \rightarrow P ^ {N} $ of the product $ P ^ {n} \times P ^ {m} $ of projective spaces into the projective space $ P ^ {N} $, where $ N = nm + n + m $. If $ x = ( u _ {0} : \dots : u _ {n} ) \in P ^ {n} $, $ y = ( v _ {0} : \dots : v _ {m} ) \in P ^ {m} $, and $ w _ {i,j} $( $ i = 0 \dots n $; $ j = 0 \dots m $) are homogeneous coordinates in $ P ^ {N} $, then the mapping is defined by the formula:

$$ \phi ( x , y) = ( w _ {i,j} ) \in P ^ {N} , $$

where $ w _ {i,j} = u _ {i} v _ {j} $. The mapping $ \phi $ is well-defined and is a closed imbedding. The image $ \phi ( P ^ {n} \times P ^ {m} ) $ of a Segre imbedding is called a Segre variety. The case when $ n = m = 1 $ has a simple geometrical meaning: $ \phi ( P ^ {1} \times P ^ {1} ) $ is the non-singular quadric in $ P ^ {3} $ with equation $ w _ {11} w _ {00} = w _ {01} w _ {10} $. The images $ \phi ( x \times P ^ {1} ) $ and $ \phi ( P ^ {1} \times y) $ give two families of generating lines of the quadric.

The terminology is in honour of B. Segre.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001

Comments

References

[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Segre imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_imbedding&oldid=23973
This article was adapted from an original article by Val.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article