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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

##### MSC:
 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
##### Keywords:
curves; Hilbert scheme; liaison class; deformation
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##### References:
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