By David Eisenbud and Joseph Harris
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26. Let π : B → P n be the blowup of P n at a point p, and let E = π −1 (p) be the exceptional divisor. Let E0 ⊂ · · · ⊂ En−1 = E be a complete flag of linear subspaces of E ∼ = P n−1 , and let P1 ⊂ · · · ⊂ Pn = P n n be a flag of linear subspaces of P not containing p, with dim Pi = i. (a) The 2n effective classes π ∗ [Pn−i ] = ζ i , for i = 0, . . n − 1; and [En−i ] = (−1)i−1 i , for i = 1, . . n form a free basis of the abelian group A(B). The class of the strict transform of a hyperplane H with p ⊂ H ⊂ P n is σ = ζ − .
But there is a way to “fix” both these problems. 33. There is a ring homomorphism Ch from the Grothendieck group of X to the Chow Chern character of E given by a power series in the Chern classes, c1 (E)2 − 2c2 (E) + · · · ∈ Q ⊗ A(X), 2 The Chern character is characterized by the fact that if E has a filtration 0 ⊂ E1 ⊂ · · · ⊂ Er = E, where Ei is a sub-bundle of rank i, and we set Li = Ei /Ei+1 , then Ch(E) = rank(E) + c1 (E) + r ec1 (Li ) . Ch(E) = i=1 If X is a smooth projective variety, then the map Ch : Q ⊗ K(X) → Q ⊗ A(X) is an isomorphism of rings.
We will almost always compute the canonical class of a variety X from the canonical class of some variety containing X, or in any case receiving a map from X. This gives special importance to the canonical class of projective space, which we will now compute. ). 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 .