 Lefschetz pencil

On a closed oriented smooth four-manifold $X$, a Lefschetz pencil is a smooth map $f:X\setminus B\to {\mathbb{CP}}^1$ with finite base locus $B$ and isolated critical points. Orientation-compatible complex local coordinates give the modelsA regular fiber closes to a surface through the base points. Blowing up those points resolves the local ratio and gives a Lefschetz fibration, with an exceptional section for each base point.^[ref]

For a pencil generated by homogeneous polynomials $F_0,F_1$ on a smooth projective surface, $f=[F_0:F_1]$ and $B$ is their common zero set. Generic choices in sufficiently positive linear systems give the isolated complex Morse singularities required here. The local models are stronger conditions than merely having a rational map to the projective line.