By David Eisenbud and Joe Harris

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**Sample text**

Using multiplicities we can extend this formula to the case where the varieties intersect properly (that is, the Zi all have codimension equal to ci ); and if we assume as well that the varieties are generically CohenMacaulay along their intersections, then the multiplicities are equal to the lengths of the components of the intersection scheme, and thus are already encoded in the intersection cycle. 2 Chow Ring Examples 27 for example when two plane curves meet along a finite set of points, the classical case treated by B´ezout’s Theorem.

A cubic form F on P 3 , through its restriction to each VL , defines an algebraic global section σF of this vector bundle. Thus the number of lines contained in the cubic surface X is the number of zeros of the section σF , and if this number is finite then at least the scheme of zeros will have multiplicity c4 (V). 2, that deg c4 (V) = 27. 34 one still has to show that the number of lines on any smooth cubic surface is finite, and that the zeros all occur with multiplicity 1. This whole series of steps will be carried out in detail in Chapter 8.

We ˜ k = α−1 (Γ can write the open stratum Γ k−1 \ Γk−2 ) ∩ (B \ Λ) as the locus ˜ k = {((1, 0, . . , 0, λ, λyn−k+2 , . . , λyn ), (0, . . , 0, 1, yn−k+2 , . . , yn ))} ; Γ ˜ k with A k . the functions λ, yn−k+2 , . . , yn then give an isomorphism of Γ It follows that the classes λk = [Λk ] and γk = [Γk ] ∈ Ak (B) generate the Chow groups of B; we now want to say what their intersection products are. Since Λk is the the preimage of a k-plane in P n not containing p, and any two such are linearly equivalent in P n , the class of the pullback of any k-plane in P n not containing p is also equal to λk .