 Lefschetz pencil

A Lefschetz pencil on an oriented four-manifold is a map $f:X\setminus B\to {\mathbb{CP}}^1$ with a finite base locus $B$, locally modeled by $[z_1:z_2]$ at each base point, and isolated critical points locally modeled by $z_1^2+z_2^2$. Blowing up the base points turns the pencil into a Lefschetz fibration.