# 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