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 $\mathbb CP^1$.

iii)The problem fibers, i.e., the fibers at non-regular points are "fishtail fibers", i.e., vanishing cycles, and they are finitely-many, and these are described by their monodromy. The monodromy is described by a Dehn twists at each fiber, and the fibration can be fully described as a word (in the mapping class group of the fiber, given by the Dehn twists) given and the entire fibration can be fully described by this word

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.

The Topology of Complex Projective Varieties after S.Lefschetz. $\endgroup$