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The curve \({\tilde{\mathcal C}}_ 4=(\lambda^ 4,\lambda^ 3\mu,\lambda\mu^ 3,\mu^ 4)\subset {\mathbb{P}}^ 3_ k\), is not set- theoretic complete intersection of two quartic surfaces. (English) Zbl 0613.14025
This paper shows that the Cremona quartic curve \(\{t^ 4,t^ 3s,ts^ 3,s^ 4\}\) is not set-theoretically a complete intersection of two quartic surfaces if the characteristic of the base field (algebraically closed) is different from 2 and 3. This may shed some light on the general problem whether a projective space curve is always set- theoretically a complete intersection. Note that the first author [Rend. Semin. Mat. Univ. Padova 65, 177-190 (1981; Zbl 0492.14020)] has proven that this curve is not set-theoretically a complete intersection of any pair of surfaces of degrees 3 and 4 and that this result has been generalized by E. Stagnaro [Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 7, No.1, 51-87 (1983; Zbl 0545.14025)] to any rational non-singular quartic curve.
Reviewer: Ngo Viet Trung

MSC:
14H45 Special algebraic curves and curves of low genus
14N05 Projective techniques in algebraic geometry
14M10 Complete intersections
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References:
[1] P.C. Craighero , Una osservazione sulla curva di Cremona di P k3: (\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
[2] E. Stagnaro , Sulle curve razionali non singolari di ordine 4 di Pk3 , Rend. Acc. Naz. Scienze dei XL (memorie di matematica) , 104 , VII , 6 ( 1983 ), pp. 51 - 88 .
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