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No rational nonsingular quartic curve \({\mathcal C}_ 4\subset {\mathbb{P}}^ 3\) can be set-theoretic complete intersection on a cubic surface. (English) Zbl 0706.14031
Generalizing a result of E. Stagnaro [Rend. Accad. Naz. Sci. Detta XL, V. Ser. Mem. Mat. 7, No.1, 51-87 (1983; Zbl 0545.14025)], the authors show that smooth rational quartic curves in \({\mathbb{P}}_ k^ 3\) (k algebraically closed of \(char(k)=0)\) are never set-theoretic complete intersections of any pairs of surfaces \(F_ 3\) and \(F_ n\), for every degree n.
Reviewer: M.Herrmann

MSC:
14M10 Complete intersections
14H50 Plane and space curves
14M20 Rational and unirational varieties
Citations:
Zbl 0545.14025
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References:
[1] R. Hartshorne , Algebraic Geometry , Springer , Berlin ( 1977 ). MR 463157 | Zbl 0367.14001 · Zbl 0367.14001
[2] P.C. Craighero , Una osservazione sulla curva di Cremona di P3: C = (\lambda \mu 3, \lambda 3\mu , \lambda 4, \mu 4) , Rend. Sem. Mat. Univ. Padova , 65 ( 1981 ), pp. 177 - 190 . Numdam | Zbl 0492.14020 · Zbl 0492.14020
[3] P.C. Craighero - R. Gattazzo , The curve C = (\lambda 4, \lambda 3\mu , \lambda \mu 3, \mu 4) \subset P3 is not set-theoretic complete intersection of two quartic surfaces , Rend. Sem. Mat. Univ. Padova , 76 ( 1986 ), pp. 177 - 200 . Numdam | Zbl 0613.14025 · Zbl 0613.14025
[4] D. Gallarati , Appunti di Geometria, Di Stefano , Genova ( 1969 ).
[5] D.B. Jaffe , Space curves which are the intersection of a cone with another surface , to appear in Duke Math. Journal. Article | MR 975125 | Zbl 0677.14011 · Zbl 0677.14011
[6] E. Kunz , Einfuhrung in die Kommutative Algebra und Algebraische Geometrie , vieweg , Wiesbaden ( 1979 ). MR 562105 | Zbl 0432.13001 · Zbl 0432.13001
[7] L. Robbiano , Some properties of complete intersection in good projective varieties , Nagoya Math. J. , 61 ( 1976 ), pp. 103 - 111 . Article | MR 409467 | Zbl 0309.14043 · Zbl 0309.14043
[8] E. Stagnaro , Le superficie cubiche di P3 a curve sottoinsieme intersezione completa , Atti Acc. Ligure Sc. Arti , 36 ( 1975 ), pp. 138 - 148 . MR 360606 | Zbl 0363.14014 · Zbl 0363.14014
[9] E. Stagnaro , Sulle curve razionali non singolari di ordine 4 di P3 , Rend. Acc. Naz. Scienze detta dei XL, Memorie di Matematica , 101 , ( 6 ) VII ( 1983 ), pp. 51 - 88 . Zbl 0545.14025 · Zbl 0545.14025
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