By David Eisenbud and Joseph Harris

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18. A(U0 ) is generated by [U0 ]. By part b) of Proposition ?? the sequence Z · [U0 ] = A(U0 ) ✲ A(X) ✲ A(X \ U0 ) ✲ 0. is right exact. Since the classes in A(U ) of the closed strata in U come from the classes of (the same) closed strata in X, it follows that A(X) is generated by the classes of the closed strata. 20. The Chow ring of P n is A∗ (P n ) = Z[ζ]/(ζ n+1 ), and the class of a variety of codimension k and degree d is dζ k . Proof. 19 that the Chow group Ak (P n ) of P n is generated by the class of any k-plane Lk ⊂ P n .

6 Tangent Bundle, Canonical Class and Adjunction 39 then the condition that L1 ∩ L2 ∩ L3 = ∅ is equivalent to the vanishing a1,1 a2,1 a3,1 a1,2 a2,2 a3,2 a1,3 a2,3 = 0. a3,3 This is a homogeneous trilinear form on P 2 × P 2 × P 2 , from which we see that Φ is indeed a closed subvariety of P 2 × P 2 × P 2 , and A is likewise a closed subvariety of P 9 . Moreover, we see that the class of Φ is [Φ] = α1 + α2 + α3 ∈ A1 (P 2 × P 2 × P 2 ), so that the pullback via µ of 5 general hyperplanes in P 9 will intersect Φ in 6 5 6 [Φ] (α1 + α2 + α3 ) = (α1 + α2 + α3 ) = = 90 2, 2, 2 points; again, since the map µ|Φ : Φ → A has degree 6, it follows that the degree of the locus A ⊂ P 9 of asterisks is 15.

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. 1 The Chow Group and the Intersection Product 21 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] sends [A] to a nonzero integer.

### 3264 & All That: A second course in algebraic geometry. by David Eisenbud and Joseph Harris

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Categories: Algebraic Geometry