Oriented intersection multiplicities
Barge and Morel defined a graded “oriented Chow group” of a smooth variety over a field, which may be viewed as a quotient of a group of “oriented algebraic cycles” modulo a suitable equivalence relation. A precursor was the idea of an oriented 0-cycle, suggested by M. Nori, which led to the Euler class group, considered in works of Raja Sridharan, S. M. Bhatwadekar, S. Mandal, and others. Fasel constructed an intersection product on the oriented Chow groups of Barge and Morel, leading to an “oriented Chow ring”, which admits a graded ring homomorphism to the “usual” Chow ring of (unoriented) cycles. In my lecture, I’ll introduce these ideas, and discuss some joint work with Fasel, about the corresponding notion of intersection multiplicities.