= Lefschetz fibration
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A Lefschetz fibration is a <proper map> from an oriented <smooth manifold> $X^{2n}$ to an oriented <surface>, with isolated critical <points> modeled in orientation-compatible complex coordinates by
$$
(z_1,\ldots,z_n)\longmapsto z_1^2+\cdots+z_n^2.
$$
Away from the critical values it is a proper <submersion>, hence a smooth <fiber bundle> by the <Ehresmann fibration theorem>. Properness is needed for this conclusion; the local critical-point model alone cannot ensure that fibers do not escape to infinity.<Lefschetz pencils, branched covers and symplectic invariants by Denis Auroux and Ivan Smith>[Section 2, Lefschetz fibrations and their local models]{ref}
In real <dimension> four, a regular fiber is a compact <surface>. Near a critical value, a circle in that surface shrinks to a node. Blowing up the base <points> of a <Lefschetz pencil> supplies a central family of examples: each exceptional <sphere> becomes a section of the resolved fibration.
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