On Bertini's theorem for fibrations by plane projective quartic curves in characteristic five

Abstract

Bertini's theorem on variable singular points may fail in positive characteristic. We construct in characteristic five a two-dimensional algebraic fibration π : T → S by plane projective quartic curves, that is pathological in the sense that all fibers are non-smooth though the total space T is smooth after restricting the base surface S to a dense open subset, and that is universal in the sense that each pathological fibration by plane projective quartics is up to birational equivalence obtained by a base extension either from the two-dimensional fibration π or from an one-dimensional pathological fibration π −1 ( C ) → C obtained by restricting the base of π to a uniquely determined curve C on S. These curves on the base surface S, which are bases of pathological fibrations, are classified in terms of invariant curves of an algebraic vector field. Among these fibrations there is just one whose general fiber admits a non-ordinary inflection point. In analogy to the Kodaira–Néron classification of special fibers of minimal fibrations by elliptic curves, we construct the minimal proper regular model of this pathological fibration, determine the structure of the bad fibers, and study the global geometry of the total space.