INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

40 MezzettiNote that, by definition, 1 ♯ O(X) = 1 X, for all affine X; moreover (v◦u) ♯ = u ♯ ◦v ♯for all u : O(Z) → O(Y ), v : O(Y ) → O(X), K–homomorphisms of affine algebraicsets: this means that also this construction is functorial.9.5. Corollary. Let X, Y be affine algebraic sets. Then X ≃ Y if and only ifO(X) ≃ O(Y ).□If X and Y are quasi–projective varieties and φ : X → Y is regular, it is notalways possible to define a comorphism K(Y ) → K(X). If f is a rational functionon Y with domf = U, it can happen that φ(X) ∩ domf = ∅, in which case f ◦ φdoes not exist. Nevertheless, if we assume that φ is dominant, i.e. φ(X) = Y ,then certainly φ(X) ∩ U ≠ ∅, hence 〈φ −1 (U), f ◦ φ〉 ∈ K(X). We obtain a K–homomorphism, which is necessarily injective, K(Y ) → K(X), also denoted φ ∗ .Note that in this case, we have: dim X ≥ dimY . As above, it is possible to checkthat, if X ≃ Y , then K(X) ≃ K(Y ), hence dimX = dim Y . Moreover, if P ∈ Xand Q = φ(P), then φ ∗ induces a map O Q,Y → O P,X , such that φ ∗ M Q,Y ⊂ M P,X .Also in this case, if φ is an isomorphism, then O Q,Y ≃ O P,X .Let now X ⊂ P n be a quasi–projective variety and φ : X → P m be a map.9.6. Proposition. φ is a morphism if and only if, for all P ∈ X, there existan open neighbourhood U P of P and n + 1 homogeneous polynomials F 0 , . . ., F mof the same degree, in K[x 0 , x 1 , . . ., x n ], such that, if Q ∈ U P , then φ(Q) =[F 0 (Q), . . ., F m (Q)]. In particular, for all Q ∈ U P , there exists an index i suchthat F i (Q) ≠ 0.Proof. “ ⇒” Let P ∈ X, Q = φ(P) and assume that Q ∈ U 0 . Then U := φ −1 (U 0 )is an open neighbourhood of P and we can consider the restriction φ| U : U → U 0 ,which is regular. Possibly after restricting U, using non–homogeneous coordinateson Y 0 , we can assume that φ| U = (F 1 /G 1 , . . ., F m /G m ), where (F 1 , G 1 ),. . ., (F m , G m ) are pairs of homogeneous polynomials of the same degree suchthat V P (G i ) ∩ U = ∅ for all index i. We can reduce the fractions F i /G i to acommon denominator F 0 , so that deg F 0 = deg F 1 = . . . = deg F m and φ| U =(F 1 /F 0 , . . ., F m /F 0 ) = [F 0 , F 1 , . . ., F m ], with F 0 (Q) ≠ 0 for Q ∈ U.“ ⇐” Possibly after restricting U P , we can assume F i (Q) ≠ 0 for all Q ∈ U Pand suitable i. Let i = 0: then φ| UP : U P → U 0 operates as follows: φ| UP (Q) =(F 1 (Q)/F 0 (Q), . . ., F m (Q)/F 0 (Q)), so it is a morphism by Proposition 9.3. Fromthis remark, one deduces that also φ is a morphism.□9.7. Examples.1. Let X ⊂ P 2 , X = V P (x 2 1 + x2 2 − x2 0 ), the projective closure of the unitary