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Irreducible families of curves with fixed cohomology. (English) Zbl 0658.14005
We study the following question: Let X, Y be curves of the projective space such that \(h^ i({\mathbb{P}}^ 3,I_ X(t))=h^ i({\mathbb{P}}^ 3,I_ Y(t))\) for every t,i. Do they belong to the same irreducible component of the Hilbert scheme? The answer is positive if they have also the same multiplicative structure on the graded modules \(\oplus_{t}H^ 1({\mathbb{P}}^ 3,I_ X(t)) \) and \(\oplus_{t}H^ 1({\mathbb{P}}^ 3,I_ Y(t)) \) (that is to say if they belong to the same liaison class) and that there is a deformation from X to Y through curves all in the same liaison class. Moreover, if we call \(H_ f\) the set of curves with an assigned cohomology (dimensionally), we show that it can be reducible.
Reviewer: G.Bolondi

14C05 Parametrization (Chow and Hilbert schemes)
14H45 Special algebraic curves and curves of low genus
14N05 Projective techniques in algebraic geometry
14D15 Formal methods and deformations in algebraic geometry
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[1] G. Bolondi andJ. C. Migliore, Classification of maximal rank curves in the Liaison classL n. Math. Ann.277, 585-603 (1987). · Zbl 0607.14015 · doi:10.1007/BF01457859
[2] G.Bolondi and J. C.Migliore, Buchsbaum Liaison classes. To appear in J. Algebra. · Zbl 0695.14022
[3] G. Bolondi, On the classification of curves linked to two skew lines. In: Proceedings of the Conference ?Algebraic Geometry? (Berlin), Texte Math.92, 38-52, Leipzig 1986. · Zbl 0626.14035
[4] G. Ellinsgrud, Sur le schéma de Hilbert des variétés de codimension 2 dans ? e à cône de Cohen-Macaulay. Ann. Sci. École Norm. Sup. (4)8, 423-431 (1975).
[5] L.Gruson et Chr.Peskine, Genre des courbes de l’espace projectif. In: Algebraic Geometry, Proceedings (Tromsø) 1977; LNM687, 31-59, Berlin-Heidelberg-New York 1978. · Zbl 0412.14011
[6] R.Lazarsfeld and P.Rao, Linkage of general curves of large degree. In: Algebraic geometry-open problems (Ravello) 1982; LNM997, 267-289, Berlin-Heidelberg-New York 1983.
[7] Chr. Peskine etL. Szpiro, Liaison des variétés algébriques. Invent. Math.26, 271-302 (1974). · Zbl 0298.14022 · doi:10.1007/BF01425554
[8] P. Rao, Liaison among curves in ?3. Invent. Math.50, 205-217 (1979). · Zbl 0406.14033 · doi:10.1007/BF01410078
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