Unformatted text preview: algebra. For any line L in K n +1 (i.e. a direct summand of R n +1 of rank 1) make sense of the equality F ( L ) = 0 , where F ∈ I . Show that there is a natural (make sense of this too) bijection between the set of Kpoints of Proj( R [ T , . . . , T n ] /I ) and the set of lines L in K n +1 such that F ( L ) = 0 , for any F ∈ I . 7. Show that the canonical projection p : P n R → Spec R deﬁnes a homomorphism of groups p * : Pic(Spec R ) → Pic( P n R ) with quotient isomorphic to Z (prove this ﬁrst in the case when R is a ﬁeld to get the idea). 8. Let S be a scheme and X = A n S → S be the aﬃne space over S . Show that the canonical homomorphism Pic S → Pic X is an isomorphism. 9. Let X = Spec( C [ X, Y, Z ]( XY + Z 2 ). Prove that Pic( X ) is a group of order 2. What is the class group of X ? 1...
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This note was uploaded on 02/24/2012 for the course MATH 632 taught by Professor Igordolgache during the Fall '08 term at University of MichiganDearborn.
 Fall '08
 IgorDolgache
 Algebra, Geometry

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