A projective variety over a field 
is an algebraic variety
that admits a closed embedding into a projective
space 
. Its homogeneous defining equations cut out a closed subscheme of that space. It is a quasiprojective
variety whose image is closed in the chosen projective embedding. For example,
projective space and smooth projective plane
curves are projective varieties.
A projective variety is proper over 
. The nonzero spaces of global sections
of positive powers of a line bundle on a projective
variety define rational maps whose image dimensions
measure its Iitaka dimension. A projective variety
over 
is compact in its analytic topology.
