On Moving Singularities in Fibrations by Algebraic Curves
Bertini’s theorem. Fibrations by nonsmooth curves. Relative Frobenius morphism. Nonconservative function fields. Regular but nonsmooth curves. Minimal models.
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 non-smooth 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 was motivated by the close relation between this phenomenon and the theory of regular but nonsmooth curves, or equivalently, nonconservative function fields in one variable. Actually, it also provided to understand the interesting effect of the relative Frobenius morphism in fibrations by nonsmooth curves. In analogy to the Kodaira-Nron classification of special fibers of minimal fibrations by elliptic curves, we also construct the minimal proper regular model of some fibrations by non-smooth projective plane quartic curves, determine the structure of the bad fibers, and study the global geometry of the total spaces.