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The variety of module structures. (English) Zbl 0715.14019
In this paper a curve C is a locally Cohen-Macaulay, generally locally complete intersection closed 1-dimensional subscheme of \({\mathbb{P}}^ 3(k)\), k algebraically closed field. Let F: \(\{\) 0,1,2\(\}\times {\mathbb{Z}}\to {\mathbb{N}}\) be a function and denote \(H_ F=\{C\quad curve:\) \(F(i,s)=h^ i({\mathbb{P}}^ 3,{\mathcal I}_ C(s))\) for all i,s\(\}\). \(H_ F\) is a locally closed subscheme of the Hilbert scheme \(H_{d,g}\) and the authors call \(H_ F\) the Hilbert scheme of curves with fixed cohomology (functions) F. If a multiplicative structure on the cohomology in dimension one is fixed, it is known [G. Bolondi, Arch. Math. 53, No.3, 300-305 (1989; Zbl 0658.14005)] that \(H_ F\) is irreducible, but in the paper under review it is proved that without this hypothesis \(H_ F\) is, in general, reducible. The proof is obtained by applying the study, performed in the first part of the paper, of the variety \({\mathcal V}\) parametrizing all possible structures of graded \(k[x_ 0,x_ 1,...,x_ n]\)-modules which are compatible with a graded k-vector space structure.
In particular it is shown that given \({\mathcal V}\) there are infinitely many cohomologies F such that the irreducible components of \(H_ F\) are in a 1-1 correspondence with those of \({\mathcal V}\).
Reviewer: A.Del Centina

MSC:
14H10 Families, moduli of curves (algebraic)
14C05 Parametrization (Chow and Hilbert schemes)
14F25 Classical real and complex (co)homology in algebraic geometry
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