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 (Q1088744)
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scientific article; zbMATH DE number 3991673
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | 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 |
scientific article; zbMATH DE number 3991673 |
Statements
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)
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1986
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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 \textit{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.
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Cremona quartic curve
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not set-theoretically a complete intersection
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0.93553185
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0.8405596
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0.8319383
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0.82965714
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0.8293561
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0.8266027
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0.8257929
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0.8225142
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