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On rational non-singular curves of degree 4 in \({\mathbb{P}}_ k^ 3\). (Italian. English summary) Zbl 0545.14025

Summary: The purpose of this paper is to prove the following theorem: let \(C_ 4\) be a rational non-singular quartic in \({\mathbb{P}}^ 3_ k\), k algebraically closed field of characteristic \(p\neq 2,3\); then it is not possible to find two surfaces \(F_ 3\), \(F_ 4\) in \({\mathbb{P}}^ 3_ k\), of degree 3,4 respectively, such that the complete intersection \(F_ 3\cdot F_ 4\) of \(F_ 3\) and \(F_ 4\) is \(3C_ 4\). - Moreover, in characteristic \(p=3\), we show that there exist \(C_ 4\), \(F_ 3\), \(F_ 4\) such that \(F_ 3\cdot F_ 4=3C_ 4\) determining a family \({\mathcal F}\) of \(C_ 4\) satisfying the above property, and we prove that, under a certain hypothesis, if \(C_ 4\) is a rational non-singular quartic in \({\mathbb{P}}^ 3_ k\) such that there exist \(F_ 3\) and \(F_ 4\) with \(F_ 3\cdot F_ 4=3C_ 4\), then the characteristic of k is \(p=3\) and \(C_ 4\) belongs to \({\mathcal F}\).

MSC:

14H45 Special algebraic curves and curves of low genus
14M07 Low codimension problems in algebraic geometry
14M10 Complete intersections
14G15 Finite ground fields in algebraic geometry
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