INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

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46 Mezzetticoincides with the union of the planes of the conics of V . This union results to bethe cubic hypersurface defined by the equation⎛det M = det ⎝ w ⎞00 w 01 w 02w 01 w 11 w 12⎠ = 0.w 02 w 12 w 22Indeed a point of P 5 , of coordinates [w ij ] belongs to the plane of a conic containedin V if and only if there exists a non-zero triple [b 0 , b 1 , b 2 ] which is solution of thehomogeneous system (*).Let X, Y be quasi–projective varieties.9.10. Definition. The rational maps from X to Y are the germs of regular mapsfrom open subsets of X to Y , i.e. equivalence classes of pairs (U, φ), where U ≠ ∅is open in X and φ : U → Y is regular, with respect to the relation: (U, φ) ∼ (V, ψ)if and only if φ| U∩V = ψ| U∩V . The following Lemma guarantees that the abovedefined relation satisfies the transitive property.9.11. Lemma. Let φ, ψ : X → Y be regular maps between quasi-projective varieties.If φ| U = ψ| U for U ⊂ X open and non–empty, then φ = ψ.Proof. Let P ∈ X and consider φ(P), ψ(P) ∈ Y . There exists a hyperplane Hsuch that φ(P) ∉ H and ψ(P) ∉ H. Up to a projective transformation, we canassume that H = V P (x 0 ), so φ(P), ψ(P) ∈ U 0 . Set V = φ −1 (U 0 ) ∩ ψ −1 (U 0 ): anopen neighbourhood of P. Consider the restrictions of φ and ψ from V to Y ∩U 0 :they are regular maps which coincide on V ∩ U, hence their coordinates φ i , ψ i ,i = 1, . . ., n, coincide on V ∩ U. So φ i | V = ψ i | V . In particular φ(P) = ψ(P). □A rational map from X to Y will be denoted φ : X Y . As for rationalfunctions, the domain of definition of φ, dom φ, is the maximum open subset ofX such that φ is regular at the points of dom φ.The following proposition follows from the characterization of rational functionson affine varieties.9.12. Proposition. Let X, Y be affine algebraic sets, with Y closed in A n . Thenφ : X Y is a rational map if and only if φ = (φ 1 , . . ., φ n ), where φ 1 , . . ., φ n ∈K(X).□If X ⊂ P n , Y ⊂ P m , then a rational map X Y is assigned by giving m+1homogeneous polynomials of K[x 0 , x 1 , . . ., x n ] of the same degree, F 0 , . . ., F m , suchthat at least one of them is not identically zero on X.A rational map φ : X Y is called dominant if the image of X via φ isdense in X, i.e. if φ(U) = X, where U = dom φ. If φ : X Y is dominant
Introduction to algebraic geometry 47and ψ : Y Z is any rational map, then dom ψ ∩ Imφ ≠ ∅, so we can defineψ ◦ φ : X Z: it is the germ of the map ψ ◦ φ, regular on φ −1 (dom ψ ∩ Imφ).9.13. Definition. A birational map from X to Y is a rational map φ : X Ysuch that φ is dominant and there exists ψ : Y X, a dominant rational map,such that ψ ◦ φ = 1 X and φ ◦ ψ = 1 Y as rational maps. In this case, X and Y arecalled birationally equivalent or simply birational.If φ : X Y is a dominant rational map, then we can define the comorphismφ ∗ : K(Y ) → K(X) in the usual way: it is an injective K–homomorphism.9.14. Proposition. Let X, Y be quasi–projective varieties, u : K(Y ) → K(X)be a K–homomorphism. Then there exists a rational map φ : X Y such thatφ ∗ = u.Proof. Y is covered by open affine varieties Y α , α ∈ I (by Proposition 9.9): for allindex α, K(Y ) ≃ K(Y α ) (Prop. 8.8) and K(Y α ) ≃ K(t 1 , . . ., t n ), where t 1 , . . ., t ncan be interpreted as coordinate functions on Y α . Then u(t 1 ), . . ., u(t n ) ∈ K(X)and there exists U ⊂ X, non–empty open subset such that u(t 1 ), . . ., u(t n ) are allregular on U. So u(K[t 1 , . . ., t n ]) ⊂ O(U) and we can consider the regular mapu ♯ : U → Y α ֒→ Y . The germ of u ♯ gives a rational map X Y . It is possibleto check that this rational map does not depend on the choice of Y α and U. □9.15. Theorem. Let X, Y be quasi–projective varieties. The following are equivalent:(i) X is birational to Y ;(ii) K(X) ≃ K(Y );(iii) there exist non–empty open subsets U ⊂ X and V ⊂ Y such that U ≃ V .Proof.(i) ⇔ (ii) via the construction of the comorphism φ ∗ associated to φ and ofu ♯ , associated to u : K(Y ) → K(X). One checks that both constructions arefunctorial.(i) ⇒ (iii) Let φ : X Y , ψ : Y X be inverse each other. PutU ′ = dom φ and V ′ = dom ψ. By assumption, ψ ◦ φ is defined on φ −1 (V ′ )and coincides with 1 X there. Similarly, ψ ◦ φ is defined on ψ −1 (U ′ ) and equalto 1 Y . Then φ and ψ establish an isomorphism between the corresponding setsU := φ −1 (ψ −1 (U ′ )) and V := ψ −1 (φ −1 (V ′ )).(iii) ⇒ (ii) U ≃ V implies K(U) ≃ K(V ); but K(U) ≃ K(X) and K(V ) ≃K(Y ) (Prop.8.8), so K(X) ≃ K(Y ) by transitivity.□9.16. Corollary. If X is birational to Y , then dimX = dim Y . □9.17. Examples.
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Page 41 and 42: circle. We define φ : X → P 1 by
Page 45: Then v n,d (X) ≃ X: it is the set
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