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On foliations of the real projective plane defined by decomposable pencils of cubics. (English) Zbl 1492.14009

The authors investigate pencils of cubic curves on the real projective plane \(\mathbb{R}P^2\) seen as foliations. Their main result is the following.
{Theorem 1.1.} Among regular pencils of cubics in \(\mathbb{R}P^2\), exactly the following numbers of real singular curves \(n\) and real foliation singularities \(N\) occur: \[ 0 \leq n \leq 12,\quad 1 \leq N \leq 21. \] Exactly the following numbers of centers \(c\), saddles \(sa\), and stars \(st\) occur: \[ 0 \leq c \leq 6,\quad 0 \leq sa \leq 10, \quad 0 \leq st \leq 2. \]
To prove this result they analyse the discriminant of the pencil, possible singular fibres, and the Poincaré-Hopf indices of the foliation singularities appearing in such pencils. Then they show that most cases are realised by decomposable pencils – those given by \(\alpha G_1 G_2 G_3 + \beta H_1 H_2 H_3\), for \(G_i, H_i\) linear polynomials. The remaining cases are dealt with also by exhibiting explicit examples.

MSC:

14C21 Pencils, nets, webs in algebraic geometry
14H50 Plane and space curves
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
34A26 Geometric methods in ordinary differential equations
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