By David Eisenbud and Joseph Harris

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**Sample text**

The last statement of this Theorem is the result of applying appropriate multiplicities to the set-theoretic equality f (f −1 (A) ∩ B) = A ∩ f (B) (see. 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!

Tr ) ∈ P r | f (t) = t}. Since the Fi are general, we could take them to be general translates under GLr+1 ×GLr+1 of arbitrary polynomials so the cardinality of this set is the degree of the intersection of the graph γf of f with the diagonal ∆ ⊂ P r × P r . This is δ · γf = (αr + αr−1 β + · · · + β r ) · (dr αr + dr−1 αr−1 β + · · · + β r ) = dr + dr−1 + · · · + d + 1, and the answer to the Keynote Question (the case r = d = 2) is 7. 25 implies that a general (r+1)×(r+1) matrix has r+1 eigenvalues.

We can easily write down a rational differential on P n and describe its zero and polar divisors. For example, let Z0 , . . , Zn be homogeneous coordinates on P n and zi = Zi /Z0 , i = 1, . . , n the corresponding affine coordinates on the open set U ∼ = A n where Z0 = 0, and consider the form ϕ = dz1 ∧ dz2 ∧ · · · ∧ dzn . This is visibly regular and nonzero in U so its divisor is some multiple of the hyperplane H = V (Z0 ) at infinity. To compute the multiple, let U ⊂ P n 40 1. Overture be the open set Zn = 0, and wi = Zi /Zn , i = 0, .