By David Eisenbud and Joseph Harris

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**Additional info for 3264 & All That: A second course in algebraic geometry. **

**Example text**

22 1. Overture One simple case of a projective morphism is the inclusion map from a closed subvariety ι : Y ⊂ X. When X and Y are smooth, our definition of intersections and pullbacks makes it clear that if A is any subvariety of X, then [A][Y ] is represented by the same cycle as ι∗ ([A])—except that these are considered as classes in different varieties! More precisely, we can write [A][Y ] = ι∗ (ι∗ [A]). 16 is that this cycle is well defined as a cycle on Y , not only as a cycle on X. 1 shows that it is even well-defined as a “refined intersection class” on X ∩ Y and, more generally, he proves the existence of such a refined version of the pullback under a proper, locally complete intersection morphism (of which a map of smooth projective varieties is an example).

The map π ∗ defined on cycles by π ∗ ( A ]) := π −1 (A) for every subvariety A ⊂ X preserves rational equivalence, and thus induces a map of Chow groups preserving the grading by codimension. If X and Y are smooth and quasiprojective, then π ∗ : A(X) → A(Y ) is a ring homomorphism. 1 are used. 18. If U ⊂ A n is an open set then A(U ) = An (U ) = Z·[U ]. Proof. 6, An (U ) = Z · [U ], so it suffices to show that the class [Y ] of any subvariety Y U is zero. 6, it is enough to do this in the case U = A n .

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 . 14 this shows that An (P n ) = Z. Since a general (n − k) plane intersects a general k-plane transversely in one point, multiplication by [Lk ] induces a surjective map Ak (P n ) → An (P n ) = Z, so Ak (P n ) = Z for all k. 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.

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

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