Rational elliptic surfaces with six singular double fibres

TL;DR

This paper classifies rational elliptic surfaces with a section that have exactly six singular fibers of specific types, providing multiple geometric interpretations and explicit normal forms.

Contribution

It offers a comprehensive classification of such surfaces with six singular fibers, including explicit normal forms and various geometric perspectives.

Findings

Classification of rational elliptic surfaces with six singular fibers.

Explicit normal forms for the associated plane quartic curves.

Multiple geometric interpretations of the classified surfaces.

Abstract

A rational elliptic surface with section is a smooth, rational, complex, projective surface $\mathcal{X}$ that admits a relatively minimal fibration $f: \mathcal{X}\longrightarrow \bbP^1$ such that its general fibre is a smooth irreducible curve of genus one and $f$ has a section. In this paper, we classify rational elliptic surfaces with section that have exactly six singular fibres, each counted with multiplicity two. The fibres that appear with multiplicity exactly two are either of type $II$ or of type $I_2$ of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of $\bbF_2$ branched over an appropriate trisection of the ruling of $\bbF_2$ plus the negative section, a double cover of the plane branched along a quartic curve, plus the datum of a point on the plane. Moreover, either we give explicit normal forms for the plane quartic curve, or we indicate how to find it.