affine variety

An affine varietyMathworldPlanetmath over an algebraically closed field k is a subset of some affine space kn over k which can be described as the vanishing set of finitely many polynomialsPlanetmathPlanetmath in n variables with coefficients in k, and which cannot be written as the union of two smaller such sets.

For example, the locus described by Y-X2=0 as a subset of 2 is an affine variety over the complex numbersMathworldPlanetmathPlanetmath. But the locus described by YX=0 is not (as it is the union of the loci X=0 and Y=0).

One can define a subset of affine space kn or an affine variety in kn to be closed if it is a subset defined by the vanishing set of finitely many polynomials in n variables with coefficients in k. The closed subsets then actually satisfy the requirements for closed sets in a topology, so this defines a topology on the affine variety known as the Zariski topologyMathworldPlanetmath. The definition above has the extra condition that an affine variety not be the union of two closed subsets, i.e. it is required to an irreducible topological space in the Zariski topology. Anything then satisfying the definition without possibly the irreducibility is known as an (affine) algebraic set.

A quasi-affine variety is then an open set (in the Zariski topology) of an affine variety.

Note that some geometers do not require what they call a variety to be irreduciblePlanetmathPlanetmath, that is they call algebraic sets varieties. The most prevalent definition however requires varieties to be irreducible.


  • 1 Hartshorne, Robin. Algebraic GeometryMathworldPlanetmathPlanetmath, Graduate Texts in Mathematics, Number 52. Springer-Verlag, New York, NY. 1977
Title affine variety
Canonical name AffineVariety
Date of creation 2013-03-22 12:03:55
Last modified on 2013-03-22 12:03:55
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Definition
Classification msc 14-00
Related topic ProjectiveVariety
Related topic scheme
Related topic AffineScheme
Related topic AlgebraicGeometry
Related topic AffineSpace3
Related topic Variety
Defines quasi-affine variety
Defines closed