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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