Trying to Understand Lefschetz Pencils

Question

I'm reading on Lefschetz pencils, and I'm trying to understand the following condition ii) better, although I would appreciate insights on condition i), and in general.

A Lefschetz pencil on a $4$-manifold $X$ is a pair $(B, \pi)$, where $B$ is a finite, discrete subset of $X$ , and a map $\pi$: $(X-B) > \rightarrow \mathbb CP^1 $ 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 thru $0^{2n}$ 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 holomorphic local chart in $\mathbb CP^1$.

A few questions I hope someone can help me with:

(1) There is no mention , AFAIK, of any smoothness condition for $\pi$.

Is this mention of critical point related to something else, or do we 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, 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 or references?

---

EDIT - My Background: 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.

Answer 1

Let me explain to you what inspired this definition, I think it will clarify something.

The basic example (which was originally studied by Solomon Lefschetz, as far as I understand) is the following: Consider two (smooth, irreducible) projective complex algebraic curves $C_1$ and $C_2$ in $\mathbb{CP}^2$, which are defined as zero loci of polynomials $F_1$ and $F_2$. Consider now all the possible equations of kind $\lambda_1 F_1 + \lambda F_2 = 0, \ \lambda_i \in \mathbb{C}$, and let $X$ be the union of all the curves, defined by these equations (one can say, that this is a family of curves). Each point on $X$ lies on some curve $C_{\lambda_1, \lambda_2} =\{\lambda_1 F_1 + \lambda_2F_2=0\}$, and the following holds:

i)$\lambda_1$ and $\lambda_2$ do not vanish simultaneously (there is no sense in considering trivial equation)

ii) The curve $C_{\lambda_1, \lambda_2}$ depends only on point $[\lambda_0:\lambda_1]$ on projective line

iii) If $F_1$ and $F_2$ are generic, they intersect in $\deg{F_1}\cdot \deg F_2$ points (denote this set $B$) and any two curves of the family intersect each other at these points.

Hence, we have well-defined surjective map (it is a morphism of algebraic varieties, so it is almost everywhere smooth, even holomorphic)$$\pi \colon X\setminus B \to \mathbb{P}^1, $$ which sends a point on curve $C_{\lambda_1, \lambda_2}$ to $[\lambda_1 : \lambda_2]$.

Of course, even if two basic curves $C_{1,0}$ and $C_{0,1}$ are smooth, not all the curves in the family do. Your condition 2) means that one wants to somehow control the singularities, allowing them only to be not worse than singularities of quadratic cone.

Another examples, rather simple, but extremely important, are sections of projective surface $X \subset \mathbb{CP}^3$ by a family of planes which pass through a given point. One also have to pick out the point of intersection of all the sections, unless there exists one, and ask for simple singularities.

Two final remarks which I'd like to made are:

1)In both examples we can't extend our map to $\mathbb{CP}^1$ on certain discrete set $B$, but it is possible to do after blowing it up

2)(the last but not the least) -- the condition of singularities being quadratic is indeed a condition about topology of singularities and it leads to beautiful and useful Picard-Lefschetz theory, which describes the topology of our family near the singular fiber.