Stability of foliations induced by rational maps (Q2655878)

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 \(2\). \textit{X. Gómez-Mont} and \textit{A. Lins Neto} [Topology 30, No.~3, 315--334 (1991; Zbl 0735.57014)] and later \textit{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.