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Unformatted text preview: BLOWING STUFF UP (DO NOT TRY TO SMUGGLE THROUGH TSA) 1. Computing Blowups Locally We fix X to be an integral noetherian affine scheme, X = Spec A . Most of what is said doesn’t require the scheme to be integral (that is, we don’t require A to be a domain), but certain ambiguities are avoided this way. Suppose that I is an ideal in A , I = ( z 1 ,...,z n ) and let π : e X → X be the blow-up of X along I (we really mean the blow-up along the ideal sheaf associated to I , but this abuse of notation is common and harmless). Let I denote the “inverse image ideal sheaf” of I (that is, I = π- 1 I ·O e X , the total transform). Exercise 1.1. Prove that e X can be covered by affine charts of the form U i = Spec A i = Spec A [ z 1 /z i ,...,z n /z i ] and show that the module Γ( U i , I ) is generated by z i ∈ A ⊆ A i Remark 1.2 . The same statement is essentially true in the non-integral case, but then one must be more careful by exactly what one means by A [ z 1 /z i ,...,z n /z i ]. In the integral case, all these elements live inside the fraction field. Exercise 1.3. Suppose that I and J are two ideals (or ideal sheaves by the same abuse of notation) on X (we still assume that X is integral, feel free to use this if it makes your life more convenient, or not). Let π 1 : X 1 → X be the blow-up of X along I . Let J 1 be the inverse image ideal sheaf of...
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- Spring '08
- Algebra, OX, Integral domain, Algebraic geometry, ideal sheaf