Projective algebraic set
A subset of points of a projective space defined over a field
that has (in homogeneous coordinates) the form
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Here is a homogeneous ideal in the polynomial ring
. (An ideal
is homogeneous if
and
, where the
are homogeneous polynomials of degree
, imply that
.)
Projective algebraic sets possess the following properties:
1) ;
2) ;
3) if , then
;
4) , where
is the radical of the ideal
(cf. Radical of an ideal).
It follows from properties 1)–3) that on the Zariski topology can be introduced. If
, then
can be uniquely represented as the intersection of homogeneous prime ideals:
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and
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In the case where is a homogeneous prime ideal, the projective algebraic set
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 |
Projective algebraic set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_algebraic_set&oldid=48315