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

By David Eisenbud and Joe Harris

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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 . Moreover, (Λ × Γ) ∩ ∆ ∼ =Λ∩Γ 32 1. Overture is a reduced point, so ci = (δ · αi β r−i ) = # ∆ ∩ (Λ × Γ) = #(Λ ∩ Γ) = 1.

4 respectively. The locus of reducible cubics. Let Γ ⊂ P 9 be the locus of reducible cubics and/or non-reduced cubics. We can describe Γ as the image of the map τ : P2 × P5 → P9 42 1. Overture from the product of the space P 2 of homogeneous linear forms and the space P 5 of homogeneous quadratic polynomials to P 9 , given simply by multiplication: ([F ], [G]) → [F G]. Inasmuch as the coefficients of the product F G are bilinear in the coefficients of F and G, the pullback τ ∗ (ζ) of the hyperplane class ζ ∈ A1 (P 9 ) is the sum τ ∗ (ζ) = α + β where α and β are the pullbacks to P 2 × P 5 of the hyperplane classes on P 2 and P 5 .

Overture A k-plane in P n is the transverse intersection of n − k hyperplanes so [Lk ] = ζ n−k , where ζ = [Ln−1 ] ∈ A1 (P n ) is the class of a hyperplane. Finally, since a subvariety X ⊂ P n of dimension k and degree d intersects a general n − kplane transversely in d points, we have deg([X]ζ k ) = d. Since deg(ζ n ) = 1, we conclude that [X] = dζ n−k . 22. Any map from P n to a quasi-projective variety of dimension < n is constant. Proof. Let ϕ : P n → X be the map If ϕ is not constant, then the preimage of a general hyperplane section of X will be disjoint from the preimage of a general point of X.

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