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Hilbert scheme of smooth space curves. (English) Zbl 0606.14003
The main result of this paper is that $$H_{d,g,n}$$, the Hilbert scheme of smooth irreducible curves of degree $$d$$ and genus $$g$$ in $${\mathbb{P}}^ n$$, is irreducible provided that $$d>((2n-3)g+n+3)/n$$. A particular case is $$n=3$$, when the condition reduces to $$d\geq g+3$$. - It is also shown that any integral space curve satisfying $$d\geq p_ a+3$$ is smoothable in $${\mathbb{P}}^ 3$$. The results on space curves were asserted by Severi but without complete proofs.
Reviewer: S.A.Strømme

MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14H10 Families, moduli of curves (algebraic)
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References:
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