A class of geometrically elliptic fibrations by plane projective quartic curves
Cesar Hilario, Karl Otto Stöhr
TL;DR
The paper analyzes fibrations by plane quartic curves of genus three with geometric genus one in characteristic two, under a single moving singularity whose Frobenius-image has base degree one, to achieve a birational classification via the generic fibre function field $K(C)$. It shows these fibrations are geometrically elliptic and are covered by elliptic fibrations via a purely inseparable base extension of exponent one, yielding fibrewise birationality but base inseparability. A concrete global model is built: a 4-parameter family of quartic fibres $Q\subset\mathbb{P}^2\times\mathbb{A}^4$ that leads to a fibration by integral quartics of geometric genus one, with a base-changed elliptic fibration $E_B\to B$ connected by a purely inseparable degree-two map on the base and birational on fibres. The pencil on a hyperbola specialization provides an explicit minimal regular model $\tilde S\to\mathbb{P}^1$ as an inseparable double cover of an elliptic fibration, with detailed fibre configurations and a complete desingularization, demonstrating the rationality of the resulting surfaces and the precise birational/singular-structure relationship between quartic and elliptic fibrations in characteristic two.
Abstract
We investigate fibrations by non-hyperelliptic curves of arithmetic genus three and geometric genus one in characteristic two. Assuming that there is only one moving singularity and that its image in the Frobenius pullback of the fibration has degree one over the base, we provide a complete classification up to birational equivalence. This relies on an in-depth analysis of the generic fibres, whose geometry we describe explicitly. We prove that these fibrations are covered by elliptic fibrations, and that the covers are birational on the fibres but purely inseparable of exponent one on the bases.