By David Eisenbud and Joseph Harris
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N−1, affine coordinates on U . We have 1 wi , i = 1, . . , n − 1 and zn = ; zi = w0 w0 so dw0 ∧ · · · ∧ dwn−1 ϕ = dz1 ∧ · · · ∧ zn = ± . w0n+1 The form ϕ thus has a pole of order n + 1 along H, and we conclude that KP n = −(n + 1)[H], or in other words ∼ OP n (−n − 1) ωP n = whence the well-known formulas c1 (ωP n ) = c1 (ΩP n ) = (−n − 1)ζ, where ζ is the class of a hyperplane. In this case we could take KP n to be minus the class of any hypersurface of degree n + 1. We can derive this formula more quickly with a little more technique.
20. The Chow ring of P n is A∗ (P n ) = Z[ζ]/(ζ n+1 ), and the class of a variety of codimension k and degree d is dζ k . Proof. 19 that the Chow group Ak (P n ) of P n is generated by the class of any k-plane Lk ⊂ P n . 14 this shows that An (P n ) = Z. Since a general (n − k) plane intersects a general k-plane transversely in one point, multiplication by [Lk ] induces a surjective map Ak (P n ) → An (P n ) = Z, so Ak (P n ) = Z for all k. A k-plane in P n is the transverse intersection of n − k hyperplanes so [Lk ] = ζ n−k , where ζ = [Ln−1 ] ∈ A1 (P n ) is the class of a hyperplane.
A(U0 ) is generated by [U0 ]. By part b) of Proposition ?? the sequence Z · [U0 ] = A(U0 ) ✲ A(X) ✲ A(X \ U0 ) ✲ 0. is right exact. Since the classes in A(U ) of the closed strata in U come from the classes of (the same) closed strata in X, it follows that A(X) is generated by the classes of the closed strata. 20. The Chow ring of P n is A∗ (P n ) = Z[ζ]/(ζ n+1 ), and the class of a variety of codimension k and degree d is dζ k . Proof. 19 that the Chow group Ak (P n ) of P n is generated by the class of any k-plane Lk ⊂ P n .