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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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[1] E. ARBARELLO and M. CORNALBA , A few remarks about the variety of irreducible plane curves of given degree and genus (Ann. E.N.S., Vol. 16, 1983 , pp. 467-488). Numdam | MR 86a:14020 | Zbl 0553.14009 · Zbl 0553.14009 · numdam:ASENS_1983_4_16_3_467_0 · eudml:82125
[2] E. ARBARELLO , M. CORNALBA , P. GRIFFITHS and J. HARRIS , Geometry of algebraic curves , Vol. I, Springer-Verlag, Berlin-Heidelberg-New York, 1984 . Zbl 0559.14017 · Zbl 0559.14017
[3] M. ARTIN , Deformation of singularities (Tata Lecture Notes, 1976 ).
[4] J. HARRIS , Curves in projective space (Sem. Math. Sup., 1982 , Presses Univ. Montréal). MR 84g:14024 | Zbl 0511.14014 · Zbl 0511.14014
[5] R. HARTSHORNE and A. HIRSCHOWITZ , Smoothing algebraic space curves , Preprint, 1984 , Nice. · Zbl 0574.14028
[6] L. GRUSON , R. LAZARSFELD and C. PESKINE , On a theorem of Castelnuovo and equations defining space curves (Invent. Math., Vol. 72, 1983 , pp. 491-506. MR 85g:14033 | Zbl 0565.14014 · Zbl 0565.14014 · doi:10.1007/BF01398398 · eudml:143031
[7] S. MORI , Projective Manifolds with ample tangent bundles (Ann. of Math., Vol. 110, 1979 , pp. 593-606). MR 81j:14010 | Zbl 0423.14006 · Zbl 0423.14006 · doi:10.2307/1971241
[8] F. SEVERI , Vorlesungen über algebraische Geometrie , E. Löffler Übersetzung, Leipzig, 1921 . JFM 48.0687.01 · JFM 48.0687.01
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