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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

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14H45 Special algebraic curves and curves of low genus
Full Text: DOI
[1] Amasaki, M., On the structure of arithmetically Buchsbaum curves inp^3, Publ. res. inst. math. sci., 20, 793-837, (1984) · Zbl 0574.14030
[2] Ballico, E.; Ellia, Ph., Generic curves of small genus inp^3 are of maximal rank, Math. ann., 264, 211-225, (1983) · Zbl 0501.14017
[3] Bolondi, G.; Migliore, J., Classification of maximal rank curves in the liaison class Ln, Math. ann., 277, 585-603, (1987) · Zbl 0607.14015
[4] Bolondi, G., Arithmetically normal sheaves, Bull. soc. math. France, 115, No. 1, 71-95, (1987) · Zbl 0639.14008
[5] Bolondi, G., Liaison and maximal rank, Boll. un. mat. ital. D (6), 5, No. 1, (1987) · Zbl 0719.14034
[6] Bolondi, G., On the classification of curves linked to two skew lines, (), 38-52
[7] Bolondi, G., Smoothing curves by reflexive sheaves, (), 797-803 · Zbl 0675.14011
[8] Bresinsky, H.; Schenzel, P.; Vogel, W., On liaison, arithmetical curves and monomial curves inp^3, J. algebra, 86, 283-301, (1984) · Zbl 0532.14016
[9] Ellia, Ph.; Fiorentini, M., Courbes arithme´tiquement Buchsbaum de l’espace projectif, Ann. univ. ferrara, 33, 89-110, (1988)
[10] Geramita, A.; Migliore, J., On the ideal of an arithmetically Buchsbaum curve, J. pure appl. algebra, 54, 215-247, (1988) · Zbl 0674.14036
[11] \scA. Geramita, P. Maroscia, and W. Vogel, A note on arithmetically Buchsbaum curves inP^3, Queen’s University Preprint No. 1983-24.
[12] Gruson, D.; Peskine, C., Genre des courbes de l’espace projectif, (), 31-59 · Zbl 0517.14007
[13] Hartshorne, R., On the classification of algebraic space curves, (), 83-112
[14] Hartshorne, R., Stable reflexive sheaves, Math. ann., 254, 121-176, (1980) · Zbl 0431.14004
[15] Hartshorne, R., Stable reflexive sheaves, 2, Invent. math., 66, 165-190, (1982) · Zbl 0519.14008
[16] Hartshorne, R.; Hitschowitz, Smoothing algebraic space curves, (), 98-131
[17] Lazarsfeld, R.; Rao, P., Linkage of general curves of large degree, (), 267-289
[18] Migliore, J., Geometric invariants for liaison of space curves, J. algebra, 99, 548-572, (1986) · Zbl 0596.14020
[19] Migliore, J., On linking double lines, Trans. amer. math. soc., 294, 177-185, (1986) · Zbl 0596.14019
[20] Migliore, J., Buchsbaum curves inp^3, (), 259-266
[21] Peskine, C.; Szpiro, L., Liaison des varie´te´s alge´briques, Invent. math., 26, 271-302, (1974) · Zbl 0298.14022
[22] Rao, P., Liaison among curves inp^3, Invent. math., 50, 205-217, (1979) · Zbl 0406.14033
[23] Schwartau, P., Liaison addition and monomial ideals, ()
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