Fibrations by nonsmooth genus three curves in characteristic three

Abstract

a b s t r a c tBertini'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.

On algebraic surfaces of general type with negative

“…But we think it can be of some interest in the understanding of genus changes. It implies immediately [24], Corollary 3.3. Definition 2.7 We call a curve S over K geometrically rational if S K is integral with normalisation isomorphic to P 1 K .…”

“…Definition 2.7 We call a curve S over K geometrically rational if S K is integral with normalisation isomorphic to P 1 K . A slightly weaker version of the next corollary can also be found in [24], Corollary 3.2.…”

On the classification of fibrations by genus two singular curves via fibrations by elliptic curves on surfaces

“…There is also a connection with a class of isolated hypersurfaces singularities which includes those having infinity Milnor number that appeared in [11]. The existence of these fibrations by non-smooth varieties also enables us to find other geometrical constructions that never occur in characteristic zero, for instance, a covering of P 2 with a family of singular, strange and non-classical curves, as observed in [23] Example 1.1.…”

“…Several papers studied the classification of fibrations by singular curves from the birational perspective as [29], [30], [23], [24] and [4]. All of them used the well known strategy -that in this context appeared at first in the work of Stöhr (see [29]) andwhich we outline below.…”