Varieties Defined by Quadratic Equations

Abstract

Listen

Translate article

First of all, let me fix my terminology and set-up. I will always be working over an algebraically closed ground field k. We will be concerned almost entirely with projective varieties over k (although many of our results generalize immediately to arbitrary projective schemes). By a projective variety, I will understand a topological space X all of whose points are closed, plus a sheaf βX of k-valued functions on X isomorphic to some subvariety of Pn for some n. By a subvariety of ℙn, I will mean the subset X ⊂ ℙn (k) defined by some homogeneous prime ideal A ⊂ k[Xo,…,Xn], with its Zariski-topology and with the sheaf βX of functions from X to k induced locally by polynomials in the affine coordinates. Note that our varieties have only k-rational points — no generic points. In this, we depart slightly from the language of schemes. Note too that a projective variety can be isomorphic to many different subvarieties of ℙn. An isomorphism of X with a subvariety of ℙn will be called an immersion of X in ℙn.