By David Eisenbud and Joseph Harris
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Additional resources for 3264 & All That: A second course in algebraic geometry.
3. Blowup of P 2 33 34 1. Overture P 1 in B that contains both P0 and a point of E1 as reduced divisors. The given classes are linearly independent: with respect to the intersection pairing (a, b) = deg ab on A(B), the dual basis element of [π −1 Pi ] is [π −1 Pn−i ] and the dual basis element of [±Ei is En−i , for for 0 < i < n, while the Ei do not meet the π −1 Pi . Of course the dual basis element of [P0 ] is [B]. The subvariety Pn−1 ∼ = P n−1 is the preimage in B of its image in P n , which we will call Hn−1 .
Putting these two ideas together we get ∧n ΩP n ⊗ ∧1 OP n = ∧n+1 (OPn+1 n (−1)) = OP n (−n − 1), as we saw before. If Y ⊂ X is a smooth subvariety of a smooth variety X then there is an inclusion of bundles TY ⊂ TX |Y and the quotient NY /X := TX |Y /TY is called the normal bundle of Y in X. The Adjunction formula expresses the canonical class of Y in terms of that of X. 28 (Adjunction Formula). If X is a smooth variety, and Y ⊂ X is a smooth subvariety of codimension c, then c ωY = ωX |Y ⊗ IY /X .
For example, suppose that X is a smooth cubic curve in P 2 . By the formula above the degree of X ∗ is 6. 2. 3 Products of projective spaces In general there is no K¨ unneth formula for the Chow rings of products of varieties. Even for a product of two smooth curves C and D of genera g, h ≥ 1 we have no algorithm for calculating A1 (C × D), and no idea at all what A2 (C × D) looks like, beyond the fact that it can’t be in any sense finite-dimensional (Mumford ). 28 1. 24. A(P r × P s ) ∼ = A(P r ) ⊗ A(P s ) as rings; or in other words, if we let α and β ∈ A1 (P r ×P s ) be the pullbacks, via the projection maps, of the hyperplane classes on P r and P s respectively, then A(P r × P s ) ∼ = Z[α, β]/(αr+1 , β s+1 ).