A Geometric Theory of the Buchsbaum-Rim Multiplicity Original Research Article

We develop a self-contained theory of a generalized Buchsbaum-Rim multiplicity based on elementary abstract algebraic geometry. We define the multiplicity as the sum of certain natural intersection numbers and recover the traditional definition in terms of the leading coefficient of the polynomial that gives the appropriate lengths. We interpret the partial sums as the multiplicities in certain closely associated cases. We prove an additivity formula in the rings and relate constancy to a generalized notion of "reductions." We establish a polar multiplicity formula and a mixed multiplicity formula, and we determine when each of the four types of multiplicities vanish. We end by generalizing the height inequality for maximal minors and a celebrated theorem of Böger.