A useful intersection theory requires more than the construction of rings of cycle classes on non-singular varieties. For example, if A and B are subvarieties of a non-singular variety X, the intersection product A · B should be an equivalence class of algebraic cycles closely related to the geometry of how A ∩ B, A and B are situated in X. Two extreme cases have been most familiar. If the intersection is proper, i.e., dim (A ∩ B) = dim A + dim B − dim X, then A · B is a linear combination of the irreducible components of A ∩ B, with coefficients the intersection multiplicities. At the other extreme, if A = B is a non-singular subvariety, the self-intersection formula says that A · B is represented by the top Chern class of the normal bundle of A in X. In each case A · B is represented by a cycle on A ∩ B, well-defined up to rational equivalence on A ∩ B. One consequence of the theory developed here is a construction of, and formulas for, the intersection product A · B as a rational equivalence class of cycles on A ∩ B, regardless of the dimensions of the components of A ∩ B. We call such classes refined intersection products. Similarly other intersection formulas such as the Giambelli-Thom-Porteous formulas for the degeneracy loci of a vector bundle homomorphism, are constructed on and related to the geometry of these loci, including the cases where the loci have excess dimensions.
KeywordsVector Bundle Irreducible Component Chern Class Intersection Product Basic Construction
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