Fibrations by curves with more than one nonsmooth point

Abstract

Bertini’s theorem on variable singular pointsmay fail in positive characteristic, as was discovered by Zariski in 1944. In fact, he found fibrations by nonsmooth curves. In this work we continue to classify this phenomenon in characteristic three by constructing the first example, rising in the literature, of fibrations with more than one nonsmooth point. Our approach has been motivated by the close relation between it and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. In analogy to the Kodaira-Neron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by nonsmooth projective plane quartic curves and determine the structure of the bad fibers.