By David Eisenbud and Joseph Harris

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Example text

Let f : Y → X be a map of smooth projective varieties. (a) There is a unique way of defining a map of groups f ∗ : Ac (X) → Ac (X) such that f ∗ ([A]) = [f −1 (A)] whenever f −1 (A) is generically reduced of codimension c. The map f ∗ is a ring homomorphism, and makes A into a contravariant functor from the category of smooth projective varieties to the category of graded rings. (b) (Push-Pull Formula) The map f∗ : A(Y ) → A(X) is a map of graded modules over the graded ring A(X). More explicitly, if α ∈ Ak (X) and β ∈ Al (Y ) then then f∗ (f ∗ α · β) = α · f∗ β ∈ Al−k (X).

H7 intersect Γ transversely—that is, the degree of Γ is the number of reducible cubics through p1 , . . , p7 . 51. Calculate the number of reducible plane cubics passing through 7 general points p1 , . . , p7 ∈ P 2 directly, by arguing that if C + L is any reducible cubic containing p1 , . . , p7 , the line L must contain exactly two. 52. 49): 50 1. Overture (a) Show that if p1 , . . , p6 ∈ P 2 are general points, then the degree of Σ is the number of triangles containing p1 , . . , p6 ; and (b) Calculate this number directly.

54. Once more, calculate the degree of A by finding the number of asterisks containing 5 general points p1 , . . , p5 ∈ P 2 . 55. Now let P 14 be the space of plane quartic curves, and let A ⊂ P 14 be the locus of sums of four concurrent lines. 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.

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