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Somos-4 and a quartic Surface in $\mathbb{RP}^{3}$

Helmut Ruhland

Abstract

The Somos-4 equation defines the sequences with this name. Looking at these sequences with an additional property we get a quartic polynomial in 4 variables. This polynomial defines a rational, projective surface in $\mathbb{RP}^{3}$. Here some generators of the subgroup of $Cr_3 (\mathbb{R})$ are determined, whose birational maps are automorphisms of the quartic surface.

Somos-4 and a quartic Surface in $\mathbb{RP}^{3}$

Abstract

The Somos-4 equation defines the sequences with this name. Looking at these sequences with an additional property we get a quartic polynomial in 4 variables. This polynomial defines a rational, projective surface in . Here some generators of the subgroup of are determined, whose birational maps are automorphisms of the quartic surface.
Paper Structure (14 sections, 37 equations)

This paper contains 14 sections, 37 equations.

Table of Contents

  1. Introduction
  2. About Somos-4 sequences and the group $T$ of transformations
  3. Somos-$n$: A representation of $T$ in the Cremona group $Cr_{n-1} (\mathbb{R})$
  4. Even and odd Somos-4 subsequences
  5. The quartic surface and its birational symmetry group
  6. Somos-2: The representation of $T$ in $Cr_1 (\mathbb{R})$
  7. Somos-3: The representation of $T$ in $Cr_2 (\mathbb{R})$ and invariant curves with its automorphisms
  8. A construction of curves invariant under $\hat{T}$
  9. Quadratic curves invariant under $\hat{T}$
  10. Quartic curves invariant under $\hat{T}$
  11. Sectic curves invariant under $\hat{T}$
  12. Octic curves invariant under $\hat{T}$
  13. Curves with degree $> 8$ invariant under $\hat{T}$
  14. Somos-5: The representation of $T$ in $Cr_4 (\mathbb{R})$