Definition 1. Let k be a field. An algebraic variety over k is a k-scheme X such that there exists a covering by a finite number of affine open subschemes Xi which are affine varieties over k, i.e. each Xi is the affine scheme associated to a finitely generated algebra over k. A projective variety over k is a projective scheme over k, i.e. a k-scheme isomorphic to Proj k[T0, . . . , Tn]/I for a homogeneous ideal I of k[T0, . . . , Tn].