Buchsbaum-Rim sheaves and their multiple sections

Details Buchsbaum-Rim sheaves and their multiple sections Buchsbaum-Rim sheaves and their multiple sections

1997-08-26

arxiv.org/abs/alg-geom/9708022v1

Mathematics

Commutative Algebra

27 pages, AMS-LaTeX

Scientific paper

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 $\psi$. 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.

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