INTRODUCTION TO ALGEBRAIC GEOMETRY Note del corso di ...

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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
Introduction to algebraic geometry 59where G i is homogeneous in the x’s. A point P(y 1 , . . ., y m ) is in p 2 (Z) if and onlyif the system{G i (x 0 , . . ., x n ; y 0 , . . ., y m ) = 0, i = 1, . . ., thas a solution in P n , i.e. if the ideal of K[x 0 , . . ., x n ] generated by G 1 (x; y),. . .,G t (x; y) has at least one zero in P n . Hencep 2 (Z) = {(y 1 , . . ., y m )| ∀ d ≥ 1 〈G 1 (x; y), . . ., G t (x; y)〉 ⊅ K[x 0 , . . ., x n ] d } == ⋂ d≥1{(y 1 , . . ., y m )| 〈G 1 (x; y), . . ., G t (x; y)〉 ⊅ K[x 0 , . . ., x n ] d }.Let {M α } α∈I be the set of the monomials of degree d in K[x 0 , . . ., x n ]; let d i =deg G i (x; y); let {N β i } be the set of the monomials of degree d − d i; let finallyT d = {(y 1 , . . ., y m )| 〈G 1 (x; y), . . ., G t (x; y)〉 ⊅ K[x 0 , . . ., x n ] d }.Then P(y 1 , . . ., y m ) ∉ T d if and only if M α = ∑ i G i(x; y)F i,α (x 0 , . . ., x n ), forall α and for suitable polynomials F i,α homogeneous of degree d −d i . So P ∉ T d ifand only if for all α M α is a linear combination of the polynomials {G i (x; y)N β i },i.e. the matrix A of the coefficients of the polynomials G i (x; y)N β i with respect tothe basis {M α } has maximal rank ( )n+dd . So Td is the set of zeroes of the minorsof a fixed order of the matrix A, hence it is closed.□12.5. Corollary. Let X be a projective variety. Then O(X) ≃ K.12.6. Corollary. Let X be a projective variety, φ : X → Y be any regular map.Then φ(X) is a projective variety. In particular, if X ≃ Y , then Y is projective.References.[1] M. Atiyah - I. MacDonald, Introduction to Commutative Algebra, Addison- Wesley (1969)[2] J. Harris, Algebraic Geometry, Springer (1992)[3] R. Hartshorne, Algebraic Geometry, Springer (1977)[4] E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry,Birkhäuser (1980)[5] W. Fulton, Algebraic Curves, Benjamin (1969)[6] S. Lang, Algebra, 2 nd ed., Addison-Wesley (1984)[7] I. Šafarevič, Basic Algebraic Geometry, Springer (1974)[8] R. Walker, Algebraic Curves, Springer (1978)[9] J. Semple - L. Roth, Algebraic Geometry, 1957[10] P. Samuel, O. Zariski, Commutative algebra (2 vol.), Van Nostrand, 1958[11] J. Dieudonné, Cours de géométrie algébrique, 2 / Précis de géométriealgébrique élémentaire, Presses Universitaires de France, 1974
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Page 41 and 42: circle. We define φ : X → P 1 by
Page 45 and 46: Then v n,d (X) ≃ X: it is the set
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