##### Wiktionary

**affine variety**

n. (context algebraic geometry English) A set of points (in ''n''-dimensional space) which satisfy a set of equations which have a polynomial of ''n'' variables on one side and a zero on the other side.

##### Wikipedia

**Affine variety**

In algebraic geometry, an **affine variety** over an algebraically closed field *k* is the zero-locus in the affine n-space *k* of some finite family of polynomials of *n* variables with coefficients in *k* that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) ** algebraic set**. A Zariski open subvariety of an affine variety is called a quasi-affine variety.

If *X* is an affine variety defined by a prime ideal *I*, then the quotient ring

*k*[*x*, …, *x*]/*I*

is called the **coordinate ring** of *X*. This ring is precisely the set of all regular functions on *X*; in other words, it is the space of global sections of the structure sheaf of *X*. A theorem of Serre gives a cohomological characterization of an affine variety; it says an algebraic variety is affine if and only if

*H*(*X*, *F*) = 0

for any *i* > 0 and any quasi-coherent sheaf *F* on *X*. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.

An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces, fibers of algebraic vector bundles.

An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring. In complex geometry, an affine variety is an analog of a Stein manifold.