from the total space of the line bundle the fibers over the points of $h^{-1}(\{|z_1|\le 1/2\})$ and glue $\psi({\mathbb D}\times {\mathbb D})$ to the remaining set using $H_1$. In this way we get a new holomorphic surface which contains $C$ (the same curve we started with) and a holomorphic foliation transverse to $C$ except at $p$, where the tangency is simple. Furthermore, the “local holonomy” of the new foliation at $p$ is exactly $i$. We repeat the same procedure for all critical points of $f$. At the end, we have a holomorphic surface that contains $C$ and a holomorphic foliation which is transverse to $C$ except at the critical points of $f$; we may even assume that the self-intersection number of $C$ is $n\in \mathbb N$. The map $f$ can be extended along the leaves (because of its compatibility with the involutions involved), producing the desired lifting. A similar construction can be made if the critical points are not simple.

Let us give two examples of pairs (curve, surface) which are not embeddable in the projective plane.

We have already noticed that, in order to be embeddable in the projective plane, all the submersions defined in the neighborhood of the 