2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63744We prove an extended Lefschetz principle for a large class of pencils of hypersurfaces having isolated singularities, possibly in the axis, and show that the module of vanishing cycles is generated by the images of certain variation maps.14 p, more references added, Introduction and Abstract reformulated; based on the Newton Institute preprint NI01029Algebraic GeometryAlgebraic Topology32S50, 14F45, 32S22text