Buchsbaum-Rim sheaves and their multiple sections
Abstract
This paper begins by introducing and characterizing Buchsbaum-Rim sheaves on $Z = \Proj R$ 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 $\cBf$, i.e, we consider morphisms $\psi: \cP \to \cBf$ of sheaves on
Z
dropping rank in the expected codimension, where $H^0_*(Z,\cP)$ 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.