Projective algebraic set

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

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