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Rational elliptic surfaces with six singular double fibres

Ciro Ciliberto, Antonella Grassi, Rick Miranda, Alessandro Verra, Aline Zanardini

TL;DR

This work classifies rational elliptic surfaces with section that have exactly six singular fibres counted with multiplicity two. It develops and interrelates several models—the Weierstrass form, double covers of $\mathbb{F}_2$ and $\mathbb{P}^2$, and pencils of cubics—to describe all such surfaces, distinguishing special types $(a,b)$ with $(I_2,II)$ fibres and mixed types. The authors provide explicit normal forms and moduli counts: for the $I_2$-only case, two moduli, and for the $II$-only case, three moduli; mixed cases yield two moduli, with detailed algebraic descriptions via relations among quadratics $Q_1,Q_2$ or polynomials $A,B,P,Q$. They connect Cremona equivalence, Chisini’s equianharmonic framework, and double-plane representations to obtain a cohesive picture of the geometry and parameter spaces of these surfaces, including a thorough treatment of quartic curves with nodes and bitangents. The results deliver concrete models useful for both theoretical investigations and computational implementations in algebraic geometry and related areas.

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.

Rational elliptic surfaces with six singular double fibres

TL;DR

This work classifies rational elliptic surfaces with section that have exactly six singular fibres counted with multiplicity two. It develops and interrelates several models—the Weierstrass form, double covers of and , and pencils of cubics—to describe all such surfaces, distinguishing special types with fibres and mixed types. The authors provide explicit normal forms and moduli counts: for the -only case, two moduli, and for the -only case, three moduli; mixed cases yield two moduli, with detailed algebraic descriptions via relations among quadratics or polynomials . They connect Cremona equivalence, Chisini’s equianharmonic framework, and double-plane representations to obtain a cohesive picture of the geometry and parameter spaces of these surfaces, including a thorough treatment of quartic curves with nodes and bitangents. The results deliver concrete models useful for both theoretical investigations and computational implementations in algebraic geometry and related areas.

Abstract

A rational elliptic surface with section is a smooth, rational, complex, projective surface that admits a relatively minimal fibration such that its general fibre is a smooth irreducible curve of genus one and 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 or of type of the Kodaira classification. We interpret our classification from various viewpoints: a pencil of plane cubic curves, the Weierstrass equation, a double cover of branched over an appropriate trisection of the ruling of 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.
Paper Structure (40 sections, 27 theorems, 71 equations, 1 table)

This paper contains 40 sections, 27 theorems, 71 equations, 1 table.

Table of Contents

  1. Generalities
  2. Rational elliptic surfaces
  3. Models
  4. The Weierstrass model
  5. The double cover of ${\mathbb F}_2$
  6. The double cover of ${\mathbb P}^2$
  7. The pencil of cubics
  8. The Mordell-Weil group and the Neron-Severi group of sections
  9. The height pairing
  10. Rational elliptic surfaces of special type $I_2$
  11. The basic example
  12. The double cover of ${\mathbb F}_2$ representation
  13. Back to the basic example
  14. The Weierstrass equation viewpoint
  15. The double cover of ${\mathbb P}^2$ representation
  16. ...and 25 more sections

Key Result

Lemma 2

Let $({\mathcal{X}},S_0)$ be a RESS of special type and let $(C,p)$ be associated to it. If $p \in C$ (i.e., a ramified model) then $p$ is not a flex for $C$.

Theorems & Definitions (62)

  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5: shiodaMW, (810)
  • Lemma 6
  • proof
  • ...and 52 more