Surfaces containing a family of plane curves not forming a fibration

TL;DR

This paper classifies smooth surfaces and manifolds swept out by one-dimensional families of plane curves or hypersurfaces that do not form a fibration, completing previous classification efforts.

Contribution

It provides a complete classification of such surfaces and manifolds, clarifying the structure of non-fibrating families of curves and hypersurfaces.

Findings

Classification of smooth surfaces with non-fibrating families of plane curves

Characterization of manifolds swept out by non-fibrating hypersurfaces

Extension of classification to higher-dimensional manifolds

Abstract

We complete the classification of smooth surfaces swept out by a 1-dimensional family of plane curves that do not form a fibration. As a consequence, we characterize manifolds swept out by a 1-dimensional family of hypersurfaces that do not form a fibration.