Canonical rational equivalence of intersections of divisors

Abstract

One way to define an operation in intersection theory is to define a map on the group of algebraic cycles together with a map on the group of rational equivalences which commutes with the boundary operation. Assuming the maps commute with smooth pullback, the extension of the operation to the setting of algebraic stacks is automatic. The goal of section 2 of this paper is to present the operation of intersecting with a principal Cartier divisor in this light. We then show how this operation lets us obtain a rational equivalence which is fundamental to intersection theory. A one-dimensional family of cycles on an algebraic variety always admits a unique limiting cycle, but a family of cycles over the punctured affine plane may yield different limiting cycles if one approaches the origin from different directions. An important step in the historical development of intersection theory was realizing how to prove that any two such limiting cycles are rationally equivalent. The results of section 2 yield, as a corollary, a new, explicit formula for this rational equivalence. Another important rational equivalence in intersection theory is the one that is used to demonstrate commutativity of Gysin maps associated to regularly embedded subschemes. In section 3, we exhibit a two-dimensional family of cycles such that the cycles we obtain from specializing in two different ways are precisely the ones we need to show to be rationally equivalent to obtain the commutativity result. Our explicit rational equivalence respects smooth pullback, and hence the generalization to stacks is automatic. This simplifies intersection theory on Deligne-Mumford stacks as in [8], where construction of such a rational equivalence fills the most difficult section of that important paper. Since our rational equivalence arises by considering families of cycles on a larger total space, we are able to deduce (section 4) that the rational equivalence is invariant under a certain naturally arising group action. The key observation is that we can manipulate the situation so that the group action extends to the total space. This equivariance result is used, but appears with mistaken proof, in [2], where an important new tool of modern intersection theory – the theory of virtual fundamental classes – is developed. Acknowledgement. The author would like to thank S. Bloch, W. Fulton, T. Graber, and R. Pandharipande for helpful advice and the organizers and staff of the MittagLeffler Institute for hospitality during the 1996–97 program in algebraic geometry.