3264 & All That - Intersection Theory in Algebraic Geometry by David Eisenbud and Joe Harris

By David Eisenbud and Joe Harris

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Proof. 6, An (U ) = Z · [U ], so it suffices to show that the class [Y ] of any subvariety Y U is zero. 6, it is enough to do this in the case U = A n . Let z = z1 , . . , zn be coordinates on A n and let x0 , x1 be homogeneous coordinates of P 1 . ” If f (z) = 0 is a function 24 1. Overture vanishing on Y , then f (z/t) = f ( xx10 z) is a function vanishing on tY . We let x1 f W = V ({xdeg f ( z) | f (z) vanishes on Y }) ⊂ A n × P 1 . 0 x0 The fiber of W over the point (t, 1) ∈ P 1 is obviously tY , so the restriction of W to A n × A 1 ⊂ A n × P 1 , the open set x1 = 0, is irreducible.

Any formula identity about the Chern classes of bundles that is true for a bundles that are iterated extensions of line bundles is true in general. A third point that is extremely useful in computation is that if Y ⊂ X is a subvariety and E is a bundle on X, then the pullback of c(E) to Y is equal to c(E|Y ); this follows at once (when E has enough global sections) from the description of the Chern classes as degeneracy loci, and can be proven in general by reducing to this special case. We will illustrate the use of these methods in the next section.

1 Chern Classes as Dependency Loci Although the theory of Chern classes is quite general (and we will give a formula that can serve as a general definition in Chapter 11), we will restrict ourselves here to bundles on smooth projective varieties. The advantage is that we can begin by tensoring E with OX (m) for large m, and get a bundle with lots of sections. It turns out that the Chern classes of E can be computed from those of E(m) = E ⊗ OX (m), so it will suffice to define Chern classes for bundles generated by their global sections.

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