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A subset of points of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751701.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751702.png" /> that has (in homogeneous coordinates) the form
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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/p/p075/p075170/p0751703.png" /></td> </tr></table>
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Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751704.png" /> is a homogeneous ideal in the polynomial ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751705.png" />. (An ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751706.png" /> is homogeneous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751708.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p0751709.png" /> are homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517010.png" />, imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517011.png" />.)
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A subset of points of a projective space  $  P  ^ {n} $
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defined over a field  $  k $
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that has (in homogeneous coordinates) the form
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 +
$$
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V ( I)  = \{ {( a _ {0}, \dots, a _ {n} ) \in P  ^ {n} } : {f
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( a _ {0}, \dots, a _ {n} ) = 0 \textrm{ for  any  }  f \in I } \}
 +
.
 +
$$
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Here  $  I $
 +
is a homogeneous ideal in the polynomial ring $  k [ X _ {0}, \dots, X _ {n} ] $.  
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(An ideal $  I $
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is homogeneous if $  f \in I $
 +
and $  f = \sum f _ {i} $,  
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where the $  f _ {i} $
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are homogeneous polynomials of degree $  i $,  
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imply that $  f _ {i} \in I $.)
  
 
Projective algebraic sets possess the following properties:
 
Projective algebraic sets possess the following properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517012.png" />;
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1) $  V ( \sum _ {i \in S }  I _ {i} ) = \cap _ {i \in S }  V ( I _ {i} ) $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517013.png" />;
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2) $  V ( I _ {1} \cap I _ {2} ) = V ( I _ {1} ) \cup V ( I _ {2} ) $;
  
3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517015.png" />;
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3) if $  I _ {1} \subset  I _ {2} $,  
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then $  V ( I _ {2} ) \subset  V ( I _ {1} ) $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517017.png" /> is the radical of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517018.png" /> (cf. [[Radical of an ideal|Radical of an ideal]]).
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4) $  V ( I) = V ( \sqrt I ) $,  
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where $  \sqrt I $
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is the radical of the ideal $  I $ (cf. [[Radical of an ideal|Radical of an ideal]]).
  
It follows from properties 1)–3) that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517019.png" /> the [[Zariski topology|Zariski topology]] can be introduced. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517021.png" /> can be uniquely represented as the intersection of homogeneous prime ideals:
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It follows from properties 1)–3) that on $  V ( I) $
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the [[Zariski topology|Zariski topology]] can be introduced. If $  I = \sqrt I $,  
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then $  I $
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can be uniquely represented as the intersection of homogeneous prime ideals:
  
<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/p/p075/p075170/p07517022.png" /></td> </tr></table>
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$$
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= \mathfrak B _ {1} \cap \dots \cap \mathfrak B _ {s}  $$
  
 
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/p/p075/p075170/p07517023.png" /></td> </tr></table>
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$$
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V ( I)  = V ( \mathfrak B _ {1} ) \cup \dots \cup V ( \mathfrak B _ {s} ) .
 +
$$
  
In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517024.png" /> is a homogeneous prime ideal, the projective algebraic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075170/p07517025.png" /> is called a projective variety.
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In the case where $  I $
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is a homogeneous prime ideal, the projective algebraic set $  V ( I) $
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is called a projective variety.
  
 
====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><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </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><TR><TD valign="top">[2]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''1''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a2]</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"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a2]</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 08:09, 13 July 2022


A subset of points of a projective space $ P ^ {n} $ defined over a field $ k $ that has (in homogeneous coordinates) the form

$$ V ( I) = \{ {( a _ {0}, \dots, a _ {n} ) \in P ^ {n} } : {f ( a _ {0}, \dots, a _ {n} ) = 0 \textrm{ for any } f \in I } \} . $$

Here $ I $ is a homogeneous ideal in the polynomial ring $ k [ X _ {0}, \dots, X _ {n} ] $. (An ideal $ I $ is homogeneous if $ f \in I $ and $ f = \sum f _ {i} $, where the $ f _ {i} $ are homogeneous polynomials of degree $ i $, imply that $ f _ {i} \in I $.)

Projective algebraic sets possess the following properties:

1) $ V ( \sum _ {i \in S } I _ {i} ) = \cap _ {i \in S } V ( I _ {i} ) $;

2) $ V ( I _ {1} \cap I _ {2} ) = V ( I _ {1} ) \cup V ( I _ {2} ) $;

3) if $ I _ {1} \subset I _ {2} $, then $ V ( I _ {2} ) \subset V ( I _ {1} ) $;

4) $ V ( I) = V ( \sqrt I ) $, where $ \sqrt I $ is the radical of the ideal $ I $ (cf. Radical of an ideal).

It follows from properties 1)–3) that on $ V ( I) $ the Zariski topology can be introduced. If $ I = \sqrt I $, then $ I $ can be uniquely represented as the intersection of homogeneous prime ideals:

$$ I = \mathfrak B _ {1} \cap \dots \cap \mathfrak B _ {s} $$

and

$$ V ( I) = V ( \mathfrak B _ {1} ) \cup \dots \cup V ( \mathfrak B _ {s} ) . $$

In the case where $ I $ is a homogeneous prime ideal, the projective algebraic set $ V ( I) $ is called a projective variety.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001

Comments

References

[a1] D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) MR0453732 Zbl 0356.14002
[a2] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Projective algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_algebraic_set&oldid=23932
This article was adapted from an original article by Vik.S. Kulikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article