Mixed buchsbaum-rim multiplicities

Abstract

We prove the results about mixed Buchsbaum-Rim multiplicities announced in [6, (9.10)(ii), p. 224], 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 [6] about the pure Buchsbaum-Rim multiplicities, the polar multiplicities, and so forth. Moreover, we identify the additivity theorem [6, (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 in [6], and we develop a new approach, based on the completed normal cone.