Buchsbaum–Rim Sheaves and Their Multiple Sections

Abstract

This paper begins by introducing and characterizing Buchsbaum–Rim sheaves on Z=ProjR, where R is a graded Gorenstein K-algebra. They are reflexive sheaves arising as the sheafification of kernels of sufficiently general maps between free R-modules. Then we study multiple sections of a Buchsbaum–Rim sheaf Bϕ, i.e, we consider morphisms ψ: P→Bϕ of sheaves on Z dropping rank in the expected codimension, where H0*(Z,P) is a free R-module. The main purpose of this paper is to study properties of schemes associated to the degeneracy locus S of ψ. It turns out that S is often not equidimensional. Let X denote the top-dimensional part of S. In this paper we measure the “difference” between X and S, compute their cohomology modules and describe ring-theoretic properties of their coordinate rings. Moreover, we produce graded free resolutions of X (and S) which are in general minimal. Among the applications we show how one can embed a subscheme into an arithmetically Gorenstein subscheme of the same dimension and prove that zero-loci of sections of the dual of a null correlation bundle are arithmetically Buchsbaum.