A quasiprojective variety over a field 
is an algebraic variety isomorphic to a locally
closed subscheme of some projective space 
.
Equivalently, it is an open set in an algebraic
set in projective space, with the induced
algebraic structure (The Stacks Project Authors 2026).
Every affine variety is quasiprojective, because affine space 
is the complement of a hyperplane
in projective space 
. Algebraic sets in projective space are also quasiprojective. Thus
quasiprojectivity allows both closed sets that are
varieties and their open sets. The topology
here is the Zariski topology.
For example, 
is a quasiprojective algebraic curve. A chosen
embedding of a quasiprojective variety into projective
space need not have closed image.
For a smooth complex quasi-projective variety, a smooth projective compactification with normal-crossing boundary separates the complete geometry from the removed divisor. Its logarithmic Kodaira dimension uses the canonical divisor plus that reduced boundary. The quasi-Albanese variety is a semiabelian target adapted to this open setting.
