A class of geometrically elliptic fibrations by plane projective quartic curves

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.