Intersection Theory pp 1-5 | Cite as

# Introduction

## Abstract

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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