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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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##### References:
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