Foliations with a radial Kupka set on projective spaces (Q2012222)

Consider the set \(\mathcal F(n,c):=\pi\{\omega \in H^0(\mathbb P^{n},\Omega^{1}_{\mathbb P^{n}}(c)\backslash 0|\omega\wedge d\omega=0\})\) parametrizing \(1\)-forms \(\omega\) that satisfy the Frobenius integrability condition. This is the space of normal Chern class \(c\) foliations of codimension one on \(\mathbb P^n\). A very interesting problem is to describe and to classify the irreducible components of \(\mathcal F(n,c)\), for fixed \(c\). In the present article the author deals with this problem. The main result asserts that the set \(\mathcal K(n,c,xdy-ydx)\) of holomorphic foliations of normal Chern class \(c\) and transversal type \(xdy-ydx\) is not empty if and only if \(c\) is even. Moreover, int this case, such a foliation can be written as \(\omega=fdg-gdf\), so it has a rational first integral. Furthermore, \(\mathcal K(n,c,xdy-ydx) \subset \mathcal R_{n}(\frac{c}{2},\frac{c}{2})\), a rational component of the space of foliations of normal Chern class \(c\), and the closure of \(\mathcal K(n,c,xdy-ydx)\) is precisely \(\mathcal R_{n}(\frac{c}{2},\frac{c}{2})\). It is important to observe that this result has been published in [\textit{M. Brunella}, Enseign. Math. (2) 55, No. 3--4, 227--234 (2009; Zbl 1198.32015)]. However, in the present paper the approach is different. The paper is very well written.