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
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