By David Eisenbud and Joseph Harris

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Ys , and each local ring OY,Yi has finite length, say li . We define the cycle Y associated to Y to be the formal combination i li Yi . We will next define the relation of rational equivalence between cycles. 1. Rational equivalence between two cycles ω0 and ω∞ on X bers. 27 the reader will see that the relation we define restricts to the classical notion of linear equivalence in the case of divisors on a smooth variety. To emphasize the analogy with ordinary cohomology theory we introduce a “boundary” map ∂X : Z∗ (X × P 1 ) → Z∗ (X) defined on free generators as follows: Let W be a subvariety of X × P 1 .

Now let P 14 be the space of quartic curves in P 2 . (a) Let Σ ⊂ P 14 be the closure of the space of reducible quartics. What are the irreducible components of Σ, and what are their dimensions and degrees? (b) Find the dimension and degree of the locus of totally reducible quartics (that is, quartic polynomials that factor as a product of four linear forms). 61. Use the Poincar´e-Hopf Theorem to compute the topological Euler characteristic of a smooth variety Y = X1 ∩ X2 ⊂ P n where Xi is a hypersurface of degree di .

13. If f : Y → X is a proper map of schemes, then the map f∗ : Z(Y ) → Z(X) defined above induces a map of groups f∗ : Ak (Y ) → Ak (X) for each k. 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. 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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