of $f$ (see subsection \[consa\]). We set $\tau_*:= 2 \, {{\rm deg \, }}\VV_1 / (1-1/d)$.

[(Briend-Duval [@BrDu2])]{} Let $\eta > 0$ and $l \geq 1$ be such that $\tau_* / d^l < \eta$. Let $L$ be a complex line in ${\mathbb {P}}^k$ not contained in $\VV$, and $\Delta \Subset \tilde \Delta$ be topological discs in $L \setminus \VV_l$. Then, for any $n \geq
l$, there exist $(1-\eta)d_t^n$ inverse branches $g_n$ on $\Delta$ satisfying: $${{\textsf {diam} \, }}g_n (\Delta)  \leq { \tilde c \, d^{-n/2}   \over \eta^{1/2} \  {{\textsf {mod}  \, }}(\tilde \Delta \setminus \Delta )^{1/2} },$$ where $\tilde c$ is a universal constant, and ${{\textsf {mod}  \, }}(\tilde \Delta \setminus \Delta )$ is the modulus of the annulus $\tilde \Delta \setminus \Delta$.

Let us recall the definition of the modulus (see Ahlfors book [@A], chapters 1 and 2). Let $\Lambda$ denote the family of curves joining the boundary components of $A:= \tilde \Delta
\setminus \Delta$. For any conformal metric $\rho$ on $A$, we respectively denote by ${{\textsf {area}  }}_\rho$ and by $l_\rho$ the area and the length with respect to $\rho$. We denote by ${{\textsf {conf} \,  }}(A)$ the set of conformal metrics giving finite area to $A$. The 