A projective variety over a field 
is an algebraic variety
that admits a closed embedding into a projective
space 
.
Its homogeneous defining equations describe a closed algebraic
set in that space, with its induced scheme structure.
Unlike a quasiprojective variety, it has
no deleted boundary in the chosen projective embedding. For example, projective
space and smooth plane curves are projective varieties.
Projectivity implies properness over 
. On a projective variety, sections of powers of a line
bundle define rational maps whose image dimensions
measure its Iitaka dimension. A projective variety
over 
is compact in its analytic topology.
