Rank of ordinary webs in codimension one. An effective method

We are interested by holomorphic $d$-webs $W$ of codimension one in a complex $n$-dimensional manifold $M$. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled), we proved in [CL] that their rank $\rho (W)$ is upper-bounded by a certain number ${\pi }^{\mathrm{\prime }}(n,d)($which, for $n\ge 3$, is stictly smaller than the Castelnuovo-Chern’s bound $\pi (n,d))$. In fact, denoting by $c(n,h)$ the dimension of the space of homogeneous polynomials of degree $h$ with $n$ unknowns, and by $h_0$ the integer such that $$c(n,h_0-1)<d\le c(n,h_0),$$ ${\pi }^{\mathrm{\prime }}(n,d)$ is just the first number of a decreasing sequence of positive integers $${\pi }^{\mathrm{\prime }}(n,d)={\rho }_{h_0-2}\ge {\rho }_{h_0-1}\ge \cdots \ge {\rho }_h\ge {\rho }_{h+1}\ge \cdots \ge {\rho }_{\mathrm{\infty }}=\rho (W)\ge 0$$ becoming stationary equal to $\rho (W)$ after a finite number of steps. This sequence is an interesting invariant of the web, refining the data of the only rank. The method is effective : theoretically, we can compute ${\rho }_h$ for any given $h$ ; and, as soon as two consecutive such numbers are equal ($\rho_h=\rho_{h+1},
h\geq h_0-2$),wecanconstructaholomorphicvectorbundle$R_h\to M$ofrank$\rho_h$,equippedwithatautologicalholomorphicconnection$\nabla^h$ whose curvature $K^h$ vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes. Examples will be given.