section3 - Solutions to Hartshorne 3.15, 2.14, 3.16...

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Solutions to Hartshorne 3.15, 2.14, 3.16 -Products of Varieties Jared English, June 2005 (As usual, please point out any typos or screw-ups that you find in these solutions.) After a brief discussion with Paul about part (a), I finished up problem 3.15 of Hartshorne: 3.15. The Product of Affine Varieties. Let X A n and Y A m be affine varieties. Give X × Y A n + m the induced Zariski topology. a. X × Y is irreducible. b. A ( X × Y ) = A ( X ) k A ( Y ) . c. X × Y is a product in the category of varieties. d. dim X × Y = dim X + dim Y . (a) Suppose that X × Y = Z 1 Z 2 where Z 1 and Z 2 are closed subsets. We need to show that either X × Y = Z 1 or Z 2 . I’ll outline the steps of the proof, then I’ll provide the details. 1. Let X i = { x X | x × Y Z i } , i = 1 , 2 . 2. X = X 1 X 2 . 3. Each X i is a closed subset of X . 4. Then X = X 1 or X 2 . 5. Finally, X × Y = Z 1 or Z 2 . After we prove steps (2) and (3), the irreducibility of X implies (4). Step (4) implies step (5) from the definition of the X i . Step 2. We can view sets of the form { x } × Y as fibers of the projection π : X × Y Y . For any point x X , we will show that the fiber π - 1 ( x ) = { x Y is contained in Z 1 or Z 2 ; hence, x X 1 or X 2 . Now, each fiber π - 1 ( x ) is a closed subset of X × Y , so we get a decomposition of π - 1 ( x ) into closed subsets π - 1 ( x ) = ( Z 1 Z 2 ) π - 1 ( x ) = ( Z 1 π - 1 ( x )) ( Z 2 π - 1 ( x )) . But π - 1 ( x ) = Y is irreducible, so π - 1 ( x ) = ( Z 1 π - 1 ( x )) or ( Z 2 π - 1 ( x )) . Thus, π - 1 ( x ) Z 1 or Z 2 . Step 3. Since Z 1 is closed, Z 1 is the zero set of a collection of polynomials { f i ( x,y ) } i I . We claim that X 1 is the zero set of the polynomials P = { f i ( x,y 0 ) k [ x ] } i I,y o Y . First, suppose that x Z ( P ) . This means that f i ( x,y 0 ) = 0 for every i
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This note was uploaded on 05/02/2010 for the course MATH 632 taught by Professor Staff during the Spring '08 term at University of Michigan.

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section3 - Solutions to Hartshorne 3.15, 2.14, 3.16...

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