= Lefschetz pencil
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{title2=$f:X\dashrightarrow\mathbb{CP}^1$}
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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 models
$$
f(z_1,z_2)=[z_1:z_2]\quad\text{near a base point},\qquad f(z_1,z_2)=z_1^2+z_2^2\quad\text{near a critical point}.
$$
A 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>.<Lefschetz pencils, branched covers and symplectic invariants by Denis Auroux and Ivan Smith>[Section 2, local models and conversion of pencils into fibrations]{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>.
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