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

By David Eisenbud and Joe Harris

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Extra resources for 3264 & All That - Intersection Theory in Algebraic Geometry

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To find the class δ = [∆] of the diagonal, we first observe that it is necessarily expressible as a linear combination δ = c0 αr + c1 αr−1 β + c2 αr−2 β 2 + · · · + cr β r for some c0 , . . , cr ∈ Z. We can determine the coefficients ci by taking the product of both sides of this expression with various classes of complementary codimension: specifically, if we intersect both sides with the class αi β r−i we have ci = (δ · αi β r−i ). We can evaluate the product (δ · αi β r−i ) directly: if Λ and Γ are general linear subspaces of codimension i and r − i respectively, then [Λ × Γ] = αi β r−i .

14. Suppose that K is a field. If X is a scheme proper over Spec K, then there is a map deg : A0 (X) → Z taking the class [p] of each closed point p ∈ X to the degree (κ(p) : K) of the extension of K by the residue field κ(p) of p. We will typically use this proposition together with the intersection product: if A is a k-dimensional subvariety of a smooth projective variety X and B is a k-codimensional subvariety of X such that A ∩ B is finite and nonempty, then the map Ak (X) → Z : [Z] → deg [Z][B] 22 1.

Both these questions can be answered in terms of a combination of the Chern classes of a vector bundle called the Chern character, c1 (E)2 − 2c2 (E) + · · · ∈ Q ⊗ A(X), 2 Whitney’s formula shows that the Chern classes are really defined on the classes of bundles in the Grothendieck group K(X) of X, which is the abelian group with a generator [E] for each vector bundle E on X and a relation [E] = [E ] + [E ] for each short exact sequence of vector bundles Ch(E) = rank(E) + c1 (E) + 0 → E → E → E → 0.

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


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