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Stability of foliations induced by rational maps. (English) Zbl 1208.32029

The space of codimension one foliations of degree \(d\) on the complex projective space \(\mathbb P^r\) is the algebraic set formed by \(1\)-forms \(\omega=\sum a_j dx_i\) with homogeneous degree \(d+1\) polynomials coefficients such that \(\sum a_jx_j=0\), satisfying a Frobenius integrability condition and having zero set of codimension at least \(2\). X. Gómez-Mont and A. Lins Neto [Topology 30, No. 3, 315–334 (1991; Zbl 0735.57014)] and later O. Calvo-Andrade [Bol. Soc. Bras. Mat., Nova Sér. 26, No. 1, 67–83 (1995; Zbl 0843.58001)] described some of the irreducible components of such a set.
The starting point of the paper under review is the proof that the irreducible components \(R(r,d_0,d_1)\) generated by dominant quasi-homogeneous rational maps are irreducible and generically reduced. Then, they generalize the result to higher codimension foliations. More detailed, they consider \(q\)-codimensional foliations on \(\mathbb P^r\), with \(1\leq q\leq r-2\), of degree \(d\) and show that, for \(r\geq 4\), singular foliations tangent to dominant rational maps \(\mathbb P^r\dashrightarrow \mathbb P^q\) form irreducible and generically reduced rational varieties of the space of \(q\)-codimensional foliations. They also study the associated Zariski tangent space and compute their projective degrees in many cases.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
57R30 Foliations in differential topology; geometric theory

Software:

schubert; SINGULAR
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References:

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