By David Eisenbud and Joseph Harris

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Extra resources for 3264 & All That: A second course in algebraic geometry.

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Find the degree of A. A natural generalization of the locus of asterisks would be the locus, in the space P N of hypersurfaces of degree d in P n , of cones. We will indeed be able to calculate the degree of this locus in general, but it will require more advanced techniques than we have at our disposal here. 3****. 56. Show that (in characteristic = 3) the locus Z ⊂ P 9 of triple lines is a cubic Veronese surface, and deduce that its degree is 9. 57. Let X ⊂ P 9 be the locus of cubics of the form 2L + M for L and M lines in P 2 .

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.

Pushforwards of equivalent cycles are equivalent degree of y over x, and this common value n is called the degree of the covering of f (A) by A. 12. Let f : Y → X be a proper map of schemes, and let A ⊂ X be a subvariety. (a) If f (A) has strictly lower dimension than A, then we define f∗ (A) = 0. (b) If dim f (A) = dim A, then the map f |A : A → f (A) is generically finite. If n := [K(A) : K(f (A))] is the degree of the extension of fields of rational functions, we say that f |A is generically finite of degree n, and we define f∗ (A) = n · f (A).

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