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.
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