Algebraic Variety

An algebraic variety is a generalization to dimensions of algebraic curves. More technically, an algebraic variety is a reduced scheme of finite type over a field . An algebraic variety in (or ) is defined as the set of points satisfying a system of polynomial equations for , 2, .... According to the Hilbert basis theorem, a finite number of equations suffices.

A variety is the set of common zeros to a collection of polynomials. In classical algebraic geometry, the polynomials have complex numbers for coefficients. Because of the fundamental theorem of algebra, such polynomials always have zeros. For example,

is the cone, and

is a conic section, which is a subvariety of the cone.

Actually, the cone and the conic section are examples of affine varieties because they are in affine space. A general variety is comprised of affine varieties glued together, like the coordinate charts of a manifold. The field of coefficients can be any algebraically closed field. When a variety is embedded in projective space, it is a projective algebraic variety. Also, an intrinsic variety can be thought of as an abstract object, like a manifold, independent of any particular embedding. A scheme is a generalization of a variety, which includes the possibility of replacing by any commutative ring with a unit. A further generalization is a moduli space stack.