Fibrations by nonsmooth genus three curves in characteristic three

Abstract

Bertini’s theorem on variable singular points may 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 a two-dimensional algebraic fibration by nonsmooth plane projective quartic curves, that is universal in the sense that the data about some fibrations by nonsmooth plane projective quartics are condensed in it. 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. Actually, it also provides an understanding of the interesting effect of the relative Frobenius morphism in fibrations by nonsmooth curves. In analogy to the Kodaira–Néron 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, determine the structure of the bad fibers, and study the global geometry of the total spaces.