Algebraic variety

Summary

An algebraic variety is the central object of algebraic geometry. It is a geometric space that is locally the set of solutions of a system of polynomial equations. In the classical setting over an algebraically closed field, the varieties are the affine, quasi-affine, projective and quasi-projective varieties. In modern language, a variety is an integral separated scheme of finite type over a field. This allows varieties that do not sit inside any projective space. Conventions differ between books, mainly on whether a variety must be irreducible.