Intersection Pairing of Cycles and Biextensions

We consider an intersection pairing of cycles, as well as the corresponding biextension. More specifically, we construct a pairing $L: Z^p(X)\times Z^q(X)\to Z^1(S)$ 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 $X_{\eta}$, 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 $\CH^p_{hom}(X)\times \CH_{hom}^q(X)$ by $F^{\times}$ 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.