Fibrations by plane projective rational quartic curves in characteristic two

TL;DR

This work provides a complete birational classification of fibrations by plane projective rational quartic curves in characteristic two. It reduces the problem to geometrically rational genus-$3$ curves over a function field $K$ and shows the generic fibre must lie in one of five explicit quartic families, with canonical and pseudocanonical invariants distinguishing them. Using function-field methods, Frobenius pullbacks, and the Bedoya–Stöhr algorithm, the authors construct canonical models and reveal that every such fibration is a degree-$2$ purely inseparable cover of a quasi-elliptic fibration, a phenomenon unique to characteristic two. They also introduce three universal fibrations realizing all possibilities up to base extension, prove their total spaces are uniruled, and illustrate the geometry with a detailed pencil of quartics whose fibres relate to quasi-elliptic structures. Together, these results illuminate the birational geometry of plane quartic fibrations in characteristic two and link them to quasi-elliptic geometry via Frobenius techniques.

Abstract

We give a complete classification, up to birational equivalence, of all fibrations by plane projective rational quartic curves in characteristic two.

Figures (2)

• Figure 1: Configuration of curves on $\widetilde{S}$
• Figure 2: Configuration of curves on $\widetilde{S}'$

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 2.1
• proof
• Remark 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• Theorem 2.5
• Remark 2.6
• proof