Mixed Buchsbaum--Rim Multiplicities

Abstract

We prove the results about mixed Buchsbaum--Rim multiplicities announced in (9.10)(ii) on p.224 of our recent paper [J.Alg.(1994)], including a general mixed-multiplicity formula. In addition, we identify these multiplicities as the coefficients of the ``leading form'' of the appropriate Buchsbaum-Rim polynomial in three variables, and we prove a positivity theorem. In fact, we define the multiplicities as the degrees of certain zero-dimensional ``mixed twisted'' Segre classes, and we develop an encompassing general theory of these new rational equivalence classes in all dimensions. In parallel, we develop a theory of pure ``twisted'' Segre classes, and we recover the main results in [J.Alg.(1994)] about the pure Buchsbaum--Rim multiplicities, the polar multiplicities, and so forth. Moreover, we identify the additivity theorem [J.Alg.(1994), (6.7b)(i), p.205] as giving a sort of residual-intersection formula, and we show its (somewhat unexpected) connection to the mixed-multiplicity formula. Also, we work in a more general setup than before, and we develop a new approach, based on the completed normal cone.