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On algebraic sets invariant by one-dimensional foliations on \(\mathbb{C} P(3)\). (English) Zbl 0770.57016
We consider the problem of extending the result of J.-P. Jouanolou on the density of singular holomorphic foliations on \(\mathbb{C} P(2)\) without algebraic solutions to the case of foliations by curves on \(\mathbb{C} P(3)\). We give an example of a foliation on \(\mathbb{C} P(3)\) with no invariant algebraic set (curve or surface) and prove that a dense set of foliations admits no invariant algebraic set.

57R30 Foliations in differential topology; geometric theory
34M99 Ordinary differential equations in the complex domain
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
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