All: I'm reading on Lefschetz pencils, and I'm trying to understand condition ii) below better, though I would appreciate insights on condition i), and in general. Please forgive if the presentation is somewhat-confused, since I have not found a single source including all details, so I have had to borrow from different places.

A Lefschetz pencil on a simply-connected $4$-manifold $X^4$ is a pair $(B, \pi)$, where $B$ is a finite, discrete subset of $X$ , and $\pi$ is a map $\pi$: $(X-B) \rightarrow \mathbb CP^1 $~$ S^2$ so that:

i) Each point $b$ in $B$ has an orientation-preserving local coordinate map to $(\mathbb C^2,0)$ in which $\pi$ corresponds to the projectivization map (i.e., every line through $0^{2n}$ becomes an equivalence class $[t_0:t_1]; t_0,t_1$ not both $0$, partitioning $\mathbb C^2 -0)$ , and :

ii) Every critical point of $\pi$ has an orientation-preserving chart in which $\pi(z_1,z_2)=z_1^2+z_2^2 $, for some local chart in $\mathbb CP^1$.

Now, part of my confusion has to see with the fact that some sources describe either $\pi$, or $X^4$ as being smooth, and some describe the charts as being holomorphic; other sources do not describe them this way, but do not explicitly exclude these conditions. I'm hoping someone may help me clarify the actual conditions on each of $\pi , X^4$ , and on whether the charts are holomorphic.

In addition , I have a few questions I hope someone can help me with:

1. Given that there is no mention , AFAIK, of any necessary smoothness condition for $\pi$, can the mention of critical point be related to something else? I am not aware of any definition of critical point that does not involve a condition of smoothness, or at least differentiability. Does one just assume $\pi$ is smooth, or at least differentiable, so that critical points are those where $d\pi$ does not have full rank?

2. What is the relevance of having a chart in which $\pi(z_1,z_2)$ equals $z_1^2+ z_2^2 $? I'm aware that these pencils extend, after blowing up each point of the finite, discrete point-set $B$ , into a full-blown (ha-ha) Lefschetz fibration. The blow up consists, AFAIK, of defining a tangent space at a "problem point" where this tangent space is not defined (moreover: is this an algebraic-geometric tangent space, or a differential-geometric one?) ,somehow patching all possible directions at a point by attaching a $\mathbb CP^n$ containing all directions .But I don't fully get the importance or relevance of these two conditions in ii). Any ideas, refs., please?

EDIT: I'm trying to include the little I understand about the algebraic-geometric perspective. please feel free to correct and comment, since my understanding from this perspective is pretty limited:

1') We start with a complex surface M (meaning Real 4-manifold).

2') We consider a codimension-2 , generic linear subspace $L \subset \mathbb CP^n$. Let $B:= L\cap M $ .By a dimension count (and "genericity"), $|B|=n < \infty$

3') We consider two generic codimension-1 subspaces $S^1, S^2$, generic other than they contain the linear subspace $L$. We have that $S_1,S_2$ can be represented as $V(p_0)$, $V(p_1)$ respectfully , i.e., as algebraic varieties, i.e., as the zero sets of two polynomials $p_0,p_1$ (not sure why this is possible, i.e., what guarantees we can do this.)

4') We consider the varieties associated to/ generated-by the above subspaces and respective polynomials , variety which is generated by any two points $[r_0:r_1], [s_0:s_1]$ in $\mathbb CP^1$ , i.e., the sets $V(r_0p_0+r_1p_1 )$ and $V(s_0p_0+s_1p_1)$. We show this two varieties intersect $S$ precisely at $B$, as in #2'). This intersection is independent of the choice of points $[s_0,s_1], [r_0,r_1]$ used, i.e., for any two points in $\mathbb CP^1 $ used, the associated varieties will intersect in $B$.

As you see, my understanding from this perspective is minimal, but I would love to understand it better.

Thank You.