Definition 1

Let X be a scheme over a field k. Then X is said to be an abstract variety over k if it is integral, separated, and of finite type over k. Usually we simply say X is a variety. If X is proper over k, it is said to be completePlanetmathPlanetmath. If the dimensionPlanetmathPlanetmath of X is one, then X is said to be a curve.

Some authors also require k to be algebraically closedMathworldPlanetmath, and some authors require curves to be nonsingular.

Calling X a variety would appear to conflict with the preexisting notion of an affine ( or projective variety. However, it can be shown that if k is algebraically closed, then there is an equivalence of categories between affine abstract varieties over k and affine varietiesMathworldPlanetmath over k, and another between projective abstract varieties over k and projective varieties over k.

This equivalence of categories identifies an abstract variety with the set of its k-points; this can be thought of as simply ignoring all the generic points. In the other direction, it identifies an affine variety with the prime spectrum of its coordinate ring: the variety in 𝔸n defined by the ideal


is identified with


A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanesMathworldPlanetmathPlanetmath. To see this, suppose we have a projective variety in n given by the homogeneous ideal f1,,fm. If we delete the hyperplane Xi=0, then we obtain an affine variety: let Tj=Xj/Xi; then the affine variety is the set of common zeros of


In this way, we can get n+1 overlapping affine varieties that cover our original projective variety. Using the theory of schemes, we can glue these affine varieties together to get a scheme; the result will be projective.

For more on this, see Hartshorne’s book Algebraic GeometryMathworldPlanetmathPlanetmath; see the bibliography for algebraic geometry for more resources.

Title variety
Canonical name Variety
Date of creation 2013-03-22 14:16:43
Last modified on 2013-03-22 14:16:43
Owner mps (409)
Last modified by mps (409)
Numerical id 9
Author mps (409)
Entry type Definition
Classification msc 14-00
Synonym abstract variety
Related topic Scheme
Related topic AffineVariety
Related topic ProjectiveVariety
Defines complete
Defines curve