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Additional
projective variety
English Wikipedia - The Free Encyclopedia
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Projective variety
In
algebraic geometry
, a
projective variety
over an
algebraically closed field
k
is a subset of some
projective n-space
P
n
over
k
that is the zero-locus of some finite family of
homogeneous polynomials
of
n
+ 1 variables with coefficients in
k
, that generate a
prime ideal
, the defining ideal of the variety. If the condition of generating a prime ideal is removed, such a set is called a
projective algebraic set
. Equivalently, an
algebraic variety
is projective if it can be embedded as a
Zariski closed
subvariety
of
P
n
. A Zariski open subvariety of a projective variety is called a
quasi-projective variety
.
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