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On the existence of the algebraic curves in projective n-space. (English) Zbl 0632.14029

We prove that for any \(n\geq 3\) and all pairs \((d,g)\) of integers (d\(\geq 1\), \(g\geq 0)\) such that \(g\leq (d-n)^ 2/4(n-1)\) there exists a smooth, irreducible nondegenerate algebraic curve in \({\mathbb{P}}^ n({\mathbb{C}})\) of degree \(d\) and genus \(g.\) Moreover, we prove the existence of such curves in \({\mathbb{P}}^ 4({\mathbb{C}})\) and \({\mathbb{P}}^ 5({\mathbb{C}})\) for all pairs (d,g) such that \(g\leq \pi _ 1(d,4)\) and \(g\leq \pi _ 1(d,5)\), respectively \([\pi _ 1(d,n)\) was introduced by J. Harris in “Curves in projective space”, Semin. Math. Super. 85 (1982; Zbl 0511.14014); p. 116].
Reviewer: O.Păsărescu

MSC:

14H45 Special algebraic curves and curves of low genus
14N05 Projective techniques in algebraic geometry

Keywords:

genus

Citations:

Zbl 0511.14014
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References:

[1] L. Bădescu, Algebraic Surfaces (Romanian). Edit. Acad. RSR, Bucharest 1981.
[2] M. Demazure, Surfaces de Del Pezzo. LNM777, 23–70 (1980). · Zbl 0444.14024
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