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Buchsbaum liaison classes. (English) Zbl 0695.14022
This article extends, and in a sense completes, the earlier work of the same authors [Math. Ann. 277, 585-603 (1987; Zbl 0607.14015)]. Up to a shift in degree the Hartshorne-Rao module \(M(C)=\oplus_{n}H^ 1({\mathbb{P}}^ 3,I_ C(n)) \) is an invariant under even linkage for curves C in \({\mathbb{P}}^ 3\); it is zero for the (arithmetically) Cohen-Macaulay curves, and in the Buchsbaum case it is a finite dimensional graded vector space. In this case the critical features are therefore largely numerical: a dimension sequence \((n_ 1,...,n_ t)\) (first and last entries non-zero) for all curves in the even linkage class, and for each particular curve the least n such that \(M(C)_ n\) is non-zero. The \(integer\quad t\) is called the diameter and n is the (left-most) shift; set \(N=\sum^{t}_{i=1}n_ i \). A. V. Geramita and J. C. Migliore [J. Pure Appl. Algebra 54, No.2/3, 215-247 (1988; Zbl 0674.14036)] showed that each even linkage class \(L_{n_ 1...n_ t}\) has a left-most shift, bounded below by 2N-2. The authors establish sharpness of the bound. They go on to investigate curves whose shift differs from the extreme by h, mostly under the assumption that \(h\leq t- 2\). Such curves can not be reduced and irreducible, for instance. Precise results are given for calculating the degree of C in case \(t=2\). Since it was shown earlier [Geramita and Migliore (loc. cit.)] that the Buchbaum curves of maximal rank have diameter at most two, and the ones of diameter one have been classified, the authors devote the rest of the paper to determining necessary and sufficient conditions for existence of smooth maximal rank curves in \(L_{mn}\) in terms of the numerical character of the curve as defined by Gruson and Peskine. For related results on characterization of codimension two Buchsbaum subschemes of \({\mathbb{P}}^ n\) see M. Chang [J. Differ. Geom. 31, No.2, 323-341 (1990; Zbl 0663.14034)].
Reviewer: M.Miller

MSC:
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14H45 Special algebraic curves and curves of low genus
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