Varieties Defined by Quadratic Equations

Abstract

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 β_X 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 à ⊂ k[X_o,…,X_n], with its Zariski-topology and with the sheaf β_X 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 ℙ_n. An isomorphism of X with a subvariety of ℙⁿ will be called an immersion of X in ℙⁿ.