I'm currently reading chapter $8.1$ (page 284) of Gompf and Stipsicz book about $4$-manifolds. They build a Lefschetz fibration on an arbitrary smooth projective surface $S\subset \Bbb C P^N$.
They do it by constructing a pencil first and that part is clear. Briefly, let $A$ be a (generic) complex codim $1$ linear subspace, and let $\{H_t\}_{t \in \Bbb C P^1}$ be the set of Hyperplanes in $\Bbb C P^N$ containing such $A$. Setting $F_t :=S\cap H_t$ we obtain our pencil structure. Let $B=A \cap S$ (by genericity it's a discrete set of points), and one can prove that the $H_t$'s intersect only on $B$. Therefore we get the holomorphic map $$ S \setminus B \to \Bbb C P^1$$ by associating to every $s \in S \setminus B$ the unique $t \in \Bbb C P^1$ s.t. $s \in F_t$.
By blowing up $B$ one gets a Lefschetz fibration $$ \pi \colon S' \to \Bbb C P^1$$ ($\pi$ is a non-constant holomorphic map)
My problem lies in understanding the critical values of such map $\pi$. In particular, in order to be a Lef. fibration, critical points must be discrete. The way the authors prove it is unclear.
Assume that the set of critical points of $\pi$ in $S'$ has complex codim $1$, i.e. $d\pi$ vanish everywhere along a curve $C$ in a fiber of $\pi$ (over a critical value). At any smooth point of $C$, we can find local coordinates in which $\pi$ is given by $\pi(z_1,z_2)=z_1^m$ for some $m\geq 2$. This is an absurd because if we consider an exceptional sphere in $S'$, call it $E_i$ then $C$ would intersect $E_i$ in a smooth point and we would have that $$E_i\cdot F \geq 2$$ clearly an absurd since the $E_i$ is a section.
What I don't understand is why we can find local coordinates s.t. $\pi$ is given by that polynomial, and why the intersection $E_i\cdot F \geq m$ i.e. is bounded below by the degree of such polynomial. My guess is that we can assume that the intersection takes place at $(0,0)$ and then we see that multiplicity of zero there is $m$. All the signs in this intersection should be positive.