Păsărescu, Ovidiu On the existence of the algebraic curves in projective n-space. (English) Zbl 0632.14029 Arch. Math. 51, No. 3, 255-265 (1988). 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 Cited in 2 ReviewsCited in 2 Documents MSC: 14H45 Special algebraic curves and curves of low genus 14N05 Projective techniques in algebraic geometry Keywords:genus Citations:Zbl 0511.14014 PDFBibTeX XMLCite \textit{O. Păsărescu}, Arch. Math. 51, No. 3, 255--265 (1988; Zbl 0632.14029) Full Text: DOI 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 [3] P.Griffiths and J.Harris, Principles of Algebraic Geometry. New York 1978. · Zbl 0408.14001 [4] L. Gruson etC. Peskine, Genre des courbes de l’espace projectif (II). Ann. Sci. École Norm. Sup. (4)15, 401–418 (1982). · Zbl 0517.14007 [5] R.Hartshorne, Algebraic Geometry. Berlin-Heidelberg-New York 1977. [6] R. Hartshorne, Genre des courbes algébriques dans l’espace projectif (d’après [4]). Séminaire Bourbaki, Astérisque92–93, 301–313 (1982). [7] J.Harris (with the collaboration of D.Eisenbud), Curves in Projective Space. Sém. Math. Sup. Univ. Montreal 1982. [8] S.Lang, Algebra. Reading, Massachusetts 1970. [9] O. Păsărescu, On the Classification of Singularities of Normal Gorenstein Surfaces with Ample Anticanonical Divisor (Romanian). Stud. Cerc. Mat.36, 227–243 (1984). · Zbl 0555.14017 [10] J. P.Serre, A Course in Arithmetic. Berlin-Heidelberg-New York 1973. · Zbl 0256.12001 [11] I. R.Šafarevič, Basic Algebraic Geometry. Berlin-Heidelberg-New York 1974. 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.