Mixed buchsbaum-rim multiplicities

Abstract

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 zerodimensional "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.