projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pⁿ 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 Pⁿ. A Zariski open subvariety of a projective variety is called a quasi-projective variety.