Preprint C85/2009
On Moving Singularities in Fibrations by Algebraic Curves

Rodrigo Salomão

**Keywords: **
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.