Given some type of fibration on a 4-manifold X with a torus regular fiber T, we may produce a new 4-manifold X_T by performing torus surgery on T. There is a natural way to extend the fibration to X_T, but a multiple fiber (nongeneric) singularity is introduced. We construct explicit generic fibrations (with only indefinite fold singularities) in a neighborhood of this multiple fiber. As an application this gives explicit constructions of broken Lefschetz fibrations on all elliptic surfaces (e.g., the family E(n)_p,q). As part of the construction we produce generic fibrations around exceptional fibers of Seifert fibered spaces.

The author was partially supported by NSF grant DMS-1148490.