# Proof of the existence of Lefschetz Pencils.

Let $S$ be a smooth complex projective surface. A Lefschetz pencil over $S$ is a rational map (which is not a morphism) $f:S--\rightarrow\mathbb P^1_{\mathbb C}$ with the following property:

All but finitely many fibers of $f$ are smooth and the singular fibers have only an ordinary double point and no more singularities.

The finite set of points $B\subseteq S$ where $S$ is not defined is called the base locus.

I'd like to see the sketch of the proof of the following theorem:

For every non singular complex projective surface there exists a Lefschetz pencil (the base locus must be non-empty)

I need to find a particular linear system $|D|$ on $S$ such that the induced rational map on $\mathbb P^1_{\mathbb C}$ is a Lefschetz pencil; but the problem is the following: how can I choose $|D|$?

Many thanks

• What does $--\rightarrow$ mean? Is it $- \longrightarrow$ (ak. longrightarrow) or has it some special meaning? – Tacet Nov 15 '14 at 17:03
• It is the notation for a rational map. – Dubious Nov 15 '14 at 17:04
• First time I meet two $-$. However, thank you. – Tacet Nov 15 '14 at 17:10
• Probably you are right, on books there is only one $-$. But on pieces of papers I use to write a "long dashed line" so I copy the same "error" on latex. However the important thing is that I mean simply a rational function and nothing more. – Dubious Nov 15 '14 at 17:14
• This is probably something you already know but: Bertini's theorem gives you a large chunk of what you're after. I guess the interesting part is showing that the singular fibers have at worst ordinary double points, which must have to do with the fact that you're looking at a surface and not something of higher dimension. I don't know much about surface singularities though. – Derek Allums Nov 15 '14 at 17:39

The best reference I know for this is Expose XVII in SGA7 II by N.M. Katz

"Pinceaux de Lefschetz: theoreme de monodromie"

I am sure there are English texts by now (probably also written by Katz) which are even available online. Let me briefly comment on where to find the relevant results in SGA7.

Let $k$ be an algebraically closed field. The definition of a Lefschetz pencil (pinceau de Lefschetz) on a smooth proper connected $k$-scheme $X$ (with respect to a fixed embedding $X\to \mathbb P^r_k$) is given on page 215 of SGA7 II.

It is shown in Theorem 2.5.2 that, if $k$ is of characteristic zero, the embedding $X \to \mathbb P^r$ is a "Lefschetz embedding". This means precisely that you can find a Lefschetz pencil on $X$ (See also Corollary 3.2.1).

The proof is given in full detail in that text.

Edit: Another very good reference is Chapter 2 of Voisin's Hodge Theory and Complex Algebraic Geometry II

• Could you please give a link of the English version you mentioned above? I didn't find it online. – Akatsuki Jun 18 '18 at 21:18