We consider an intersection pairing of cycles, as well as the corresponding biextension. More specifically, we construct a pairing L : Zp(X) × Zq(X) → Z1(S) between all codimensions p and q cycles on a variety X of relative dimension d over a base S, both over a field F. The main question we consider is under what conditions on the codimension q cycle D on X, all rational equivalences between two codimension p cycles A and A' on X become the same rational equivalence between the divisors L(A, D) and L(A', D) on S. For cycles $D$ that are algebraically trivial on the generic fiber Xeta, the divisors on S do not depend on the rational equivalence of the codimension p cycles. Nevertheless, for numerically trivial divisors and zero cycles, the image does depend on the rational equivalence of the zero cycles. Therefore, Bloch's biextension of CHphom(X) × CHqhom(X) by F× can not be extended to the numerically trivial cycles. As a part of the proof of the numerically trivial case, we give an explicit expression of the Suslin-Voevodsky isomorphism.
|School:||The University of Chicago|
|School Location:||United States -- Illinois|
|Source:||DAI-B 81/1(E), Dissertation Abstracts International|
|Keywords:||Biextension, Cycles, Determinants, Intersection, Knudsen-Mumford, Pairing|
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