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Characterization of arithmetically Buchsbaum subschemes of codimension 2 in \(\mathbb P^ n\). (English) Zbl 0663.14034
A subscheme \(Y\subset\mathbb P^ n\) is called arithmetically Buchsbaum if \(H^ p({\mathcal I}_{Y\cap M}(*))\) has trivial module structure for any linear subspace \(M\) of dimension \(m\) and \(p=1,...,m-2\). The main purpose of this article is to prove the following theorem: Let \(Y\) be a codimension-2 subscheme of \(\mathbb P^ n\) for \(n\geq 3\). Then \(Y\) is arithmetically Buchsbaum if and only if there exists an exact sequence \[ 0\to \oplus {\mathcal O}(-a_ i)\to \oplus \ell_ j\Omega^{p_ j}(-k_ j)\oplus {\mathcal O}(-c_ s)\to {\mathcal I}_ Y\to 0 \] where \(P_ j\neq 0\).
The theorem, coupled with a general result [see the review of the author’s paper, J. Reine Angew. Math. 397, 214–219 (1989; Zbl 0663.14008)] concerning smoothness or reducedness of the dependency locus of a map between vector bundles, allows us to determine the admissible \((a_ i,\ell_ j,p_ j,k_ j,c_ s)\), hence to give a classification of codimension-2 arithmetically Buchsbaum subschemes in \(\mathbb P^ n\) [the author, J. Reine Angew. Math. 401, 101–112 (1989; Zbl 0672.14026)].
Other applications of the theorem include bounding the degree and the number of minimal generators of \(\oplus H^ 0({\mathcal I}_ Y(k))\) and the regularity.
Reviewer: M.Chang

14M07 Low codimension problems in algebraic geometry
14N05 Projective techniques in algebraic geometry
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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