Computations of Buchsbaum–Rim multiplicities

The Buchsbaum–Rim multiplicity is a generalization of the Samuel multiplicity and is defined on submodules of free modules Msubset ofF of a local Noetherian ring A such that Msubset ofmF and F/M has finite length. Let A=k[x,y](x,y) be a localization of a polynomial ring over a field. When F/M is isomorphic to a quotient of monomial ideals there is a region of the (x,y)-plane which corresponds to F/M. We wish to compute Buchsbaum–Rim multiplicity using the areas of pieces of this region in a manner similar to that used to compute the Samuel multiplicity of a monomial ideal. We carry out these computations in the case where F has rank 2 and F/Mcongruent withI/J where I and J are monomial ideals, with the further restriction that I is generated by two elements and J is generated by at most three elements. We find that the Buchsbaum–Rim multiplicity is at most the difference of the Samuel multiplicities of J and I with equality often holding. When equality does not hold the Buchsbaum–Rim multiplicity is the difference of the Samuel multiplicities minus a term that can be expressed in terms of areas.