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Fibrations by plane quartic curves with a canonical moving singularity

Cesar Hilario, Karl-Otto Stöhr

TL;DR

The paper classifies fibrations by integral plane projective rational quartic curves with a canonical moving singularity, showing such fibrations exist only in characteristic two. It develops a framework using quasi-elliptic function fields and their Frobenius pullbacks to realize the generic fibre as a plane quartic with a unique canonical singularity, leading to two universal fibrations in characteristic two. These universal families imply any other fibration with the same fibre properties arises via base extension, and a detailed pencil is analyzed to obtain its minimal regular model, described as a purely inseparable double cover of a quasi-elliptic fibration. In the multiplicity-three case, an explicit pencil is resolved to a relatively minimal model, revealing a complex bad fibre and reinforcing the picture that these fibrations are intimately tied to quasi-elliptic geometry in characteristic two.

Abstract

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behaviour of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.

Fibrations by plane quartic curves with a canonical moving singularity

TL;DR

The paper classifies fibrations by integral plane projective rational quartic curves with a canonical moving singularity, showing such fibrations exist only in characteristic two. It develops a framework using quasi-elliptic function fields and their Frobenius pullbacks to realize the generic fibre as a plane quartic with a unique canonical singularity, leading to two universal fibrations in characteristic two. These universal families imply any other fibration with the same fibre properties arises via base extension, and a detailed pencil is analyzed to obtain its minimal regular model, described as a purely inseparable double cover of a quasi-elliptic fibration. In the multiplicity-three case, an explicit pencil is resolved to a relatively minimal model, revealing a complex bad fibre and reinforcing the picture that these fibrations are intimately tied to quasi-elliptic geometry in characteristic two.

Abstract

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric generic fibre, which determines the generic behaviour of the special fibres, is an integral plane projective rational quartic curve over the algebraic closure of the function field of the base. It has the remarkable property that the tangent lines at the non-singular points are either all bitangents or all non-ordinary inflection tangents; moreover it is strange, that is, all the tangent lines meet in a common point. We construct two fibrations that are universal in the sense that any other fibration with the aforementioned properties can be obtained from one of them by a base extension. Furthermore, among these fibrations we choose a pencil of plane quartic curves and study in detail its geometry. We determine the corresponding minimal regular model and we describe it as a purely inseparable double covering of a quasi-elliptic fibration.
Paper Structure (6 sections, 19 theorems, 78 equations, 3 figures)

This paper contains 6 sections, 19 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quasi-elliptic function fields
  3. A singular prime that is a canonical divisor
  4. Non-hyperelliptic regular non-smooth curves
  5. Universal fibrations by rational quartics in characteristic two
  6. A moving singularity of multiplicity three

Key Result

Theorem 1.1

Let $C$ be a regular proper non-hyperelliptic geometrically integral and geometrically rational curve of arithmetic genus $h^1(C,\mathcal{O}_C)=3$ over a field $K$ of characteristic $p$. Assume in addition that $C$ admits a non-smooth point $\mathfrak{p}$ that is a canonical divisor on $C$. Then $p= Conversely, each of these equations defines a curve of the above type.

Figures (3)

  • Figure 1: Dual diagram of the bad fibre $f^*(0:1)$
  • Figure 2: Dual diagram of curves in $\widetilde{S}'$
  • Figure :

Theorems & Definitions (36)

  • Theorem 1.1: see Theorem \ref{['2023_04_10_16:50']}
  • Theorem 2.1: Queen71
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Queen71
  • Proposition 2.5
  • proof : Proof of \ref{['2024_08_10_15:42']}
  • Corollary 2.6
  • Remark 2.7
  • ...and 26 more