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Read e-book online 3264 & All That - Intersection Theory in Algebraic Geometry PDF

By David Eisenbud and Joe Harris

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The determinant of this transformation is the induced map r ∧r σ : OX = ∧r (OX ) → ∧r E. The image under ∧r σ of the global section 1 of OX is a global section of ∧r E, and if it doesn’t vanish identically, the class of its zero locus is, by definition, c1 (E). Assuming that E is generated by its global sections, we can ensure that ∧r σ(1) is not identically zero by choosing a point x in X, trivializing E in a neighborhood of this point, and choosing σ1 , . . , σr to be sections that restrict to a basis at x; then the determinant will be nonzero at x.

Xn ), (y1 , . . , yn )) | xi yj = xj yi ∀i, j ≥ 1} . Say the (k − 1)-plane Γk−1 ⊂ P n−1 is given by y1 = · · · = yn−k = 0. We ˜ k = α−1 (Γ can write the open stratum Γ k−1 \ Γk−2 ) ∩ (B \ Λ) as the locus ˜ k = {((1, 0, . . , 0, λ, λyn−k+2 , . . , λyn ), (0, . . , 0, 1, yn−k+2 , . . , yn ))} ; Γ ˜ k with A k . the functions λ, yn−k+2 , . . , yn then give an isomorphism of Γ It follows that the classes λk = [Λk ] and γk = [Γk ] ∈ Ak (B) generate the Chow groups of B; we now want to say what their intersection products are.

7. 8. 2 Chow Ring Examples 41 The locus in P 9 of curves of each type is an orbit of P GL3 and a locally closed subset of P 9 ; their closures give a stratification of P 9 . What are these loci like? What are their dimensions? Which lie in the closures of which others? Where is each one smooth and singular? What are their tangent spaces and tangent cones? What are their degrees as subvarieties of P 9 ? Some of these questions are easy to answer. , and the reader can verify them as an exercise. The inclusion relationships, also indicated in the chart, are likewise not hard; to establish that one orbit lies in the closure of another, it suffices to exhibit a one-parameter family specializing from one type to the other.

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3264 & All That - Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris

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