variety

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 complete. If the dimension of $X$ is one, then $X$ is said to be a curve.

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

Calling $X$ a variety would appear to conflict with the preexisting notion of an affine (http://planetmath.org/AffineVariety) 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 varieties 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 $\mathbb{A}^{n}$ defined by the ideal

$\left<f_{1},\ldots,f_{m}\right>$

is identified with

$\operatorname{Spec}k[X_{1},\ldots,X_{n}]/\left<f_{1},\ldots,f_{m}\right>.$

A projective variety is identified as the gluing together of the affine varieties obtained by taking the complements of hyperplanes. To see this, suppose we have a projective variety in $\mathbb{P}^{n}$ given by the homogeneous ideal $\left<f_{1},\ldots,f_{m}\right>$ . If we delete the hyperplane $X_{i}=0$ , then we obtain an affine variety: let $T_{j}=X_{j}/X_{i}$ ; then the affine variety is the set of common zeros of

$\left<f_{1}(T_{0},\ldots,T_{n}),\ldots,f_{m}(T_{0},\ldots,T_{n})\right>.$

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 Geometry; 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