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 roots to a collection of polynomials. In classical algebraic geometry, the polynomials have complex numbers for coefficients. The fundamental theorem of algebra guarantees roots for nonconstant one-variable polynomials, but a system of polynomial equations can have no common root. For example,

is the cone, and

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

The cone and the conic section are affine varieties defined by polynomial equations 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 element. A further generalization is a moduli space stack.