Ein, Lawrence Hilbert scheme of smooth space curves. (English) Zbl 0606.14003 Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469-478 (1986). 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 Cited in 14 ReviewsCited in 34 Documents MSC: 14C05 Parametrization (Chow and Hilbert schemes) 14H10 Families, moduli of curves (algebraic) Keywords:special curves; Hilbert scheme of smooth irreducible curves; space curves × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [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 [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 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.