variety (original) (raw)

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 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 hyperplanes. To see this, suppose we have a projective variety in ℙn 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.