INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

58 MezzettiProof. (i) Let Γ f ⊂ X ×Y be the graph of f: Γ f = {(x, f(x)) | x ∈ X}. It is clearthat f(X) = p 2 (Γ f ), so to prove that f(X) is closed it is enough to check that Γ f isclosed in X ×Y . Let us consider the diagonal of Y : ∆ Y = {(y, y) | y ∈ Y } ⊂ Y ×Y .If Y ⊂ P n , then ∆ Y = ∆ P n∩(Y ×Y ), so it is closed because ∆ Pn is the closed subsetdefined in Σ n,n by the equations w ij − w ji = 0, i, j = 0, . . ., n. There is a naturalmap f ×1 Y : X ×Y → Y ×Y , (x, y) → (f(x), y), such that (f ×1 Y ) −1 (∆ Y ) = Γ f .It is easy to see that f × 1 Y is regular, so Γ f is closed.Let now Z be any variety and consider p 2 : f(X) × Z → Z and the regularmap f × 1 Z : X × Z → f(X) × Z. There is a commutative diagram:X × Zp ′ 2−→Z↓ f×1 Z ր p 2f(X) × ZIf T ⊂ f(X) × Z, then (f × 1 Z ) −1 (T) is closed and p 2 (T) = p ′ 2 ((f × 1 Z) −1 (T)) isclosed because X is complete. We conclude that f(X) is complete.(ii) Let T ⊂ X be a closed subvariety and Y be any variety. We have to provethat p 2 : T × Y → Y is closed. If Z ⊂ T × Y is closed, then Z is closed also inX × Y , hence p 2 (Z) is closed because X is complete.□12.3. Corollaries.1. If X is a complete variety, then O(X) ≃ K.2. If X is an affine complete variety, then X is a point.Proof. 1. If f ∈ O(X), f can be interpreted as a regular map f : X → A 1 . ByProposition 12.2, (i), f(X) is a closed complete subvariety of A 1 , which is notcomplete. Hence f(X) is a point, so f ∈ K.2. By 1., O(X) ≃ K. But O(X) ≃ K[x 1 , . . ., x n ]/I(X), hence I(X) ismaximal. By the Nullstellensatz, X is a point.□12.4. Theorem. Let X be a projective variety. Then X is complete.Proof. (sketch, see Šafarevič [7].)1. It is enough to prove that p 2 : P n × A m → A m is closed, for all n, m. Thiscan be observed by using the local character of closedness and the affine opencoverings of quasi–projective varieties.2. If x 0 , . . ., x n are homogeneous coordinates on P n and y 1 , . . ., y m are coordinateson A m , then any closed subvariety of P n × A m can be characterized as theset of common zeroes of a set of polynomials in the variables x 0 , . . ., x n , y 1 , . . ., y m ,homogeneous in the first group of variables x 0 , . . ., x n .3. Let Z ⊂ P n × A m be closed. Then Z is the set of solutions of a system ofequations{G i (x 0 , . . ., x n ; y 1 , . . ., y m ) = 0, i = 1, . . ., t