new surface without the ordinary singularity. As for the self-intersection number $H^{\prime}\cdot H^{\prime}$, we have that $H^{\prime}\cdot H^{\prime}= H\cdot H -m(m-1)= (H\cdot H -m^2) +m$. Also any holomorphic foliation $\mathcal F$ defined in $V\cup_1^m V_j$ induces naturally a holomorphic foliation in the new surface which is $\mathcal F$ in $V$ and $\phi_j^*(\mathcal F)$ in each $D_j$. We refer to this construction as [*separating branches of $H$ at P*]{}.

![Separating branches[]{data-label="fig:viaduto"}](viaduto.png)

Let us consider then a smooth plane curve $C^{\prime}$ of degree $d^{\prime}$ and genus $g(C^{\prime})= \dfrac{(d^{\prime}-1)(d^{\prime}-2)}{2}$; it can be also immersed in the plane as a curve $C$ of degree $d$ for any $d>2g(C^{\prime})$ with a number $s$ of nodal points such that $d^2-3d-2s={d^{\prime}}^2 -3d^{\prime}$. We choose $d-d^{\prime}=k^2$ for some $k\in \mathbb N$ such that

(i) $d^{\prime}$ divides $k^3-k^2$,

(ii) $d^{\prime}$ does not divide $2k^2$;

we choose also three pencils $d\left(\dfrac {u_j}{v_j}\right)=0$ of curves of degree $k$ whose sets of $k^2$ base points lie in the regular part of $C$ and are two by two disjoint. After blowing up at these $3k^2$ points and separating branches at the nodal points of $C$ we get a curve $\tilde C$ containing in some surface with self-intersection number equal to $d^2-3k^2-2s = {d^{\prime}}^2$; 