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Dynamical systems on some elliptic modular surfaces via operators on line arrangements

Lukas Kühne, Xavier Roulleau

TL;DR

The paper investigates dynamical systems arising from operator-induced transformations on line arrangements, linking matroid realization spaces to elliptic modular surfaces. For the peculiar cases n=7 and n=8, it constructs explicit quartic models in P^3, identifies the K3 surfaces as Ξ_{1}(7) and Ξ_{1}(8), and exhibits dominant rational self-maps of degree 4 that are semi-conjugate to planar dynamics via a double-cover construction. The work combines explicit algebraic geometry with period-map techniques to realize the moduli spaces as open subschemes of modular surfaces, analyzes automorphism groups, and demonstrates the existence of periodic line-arrangement configurations. These results provide concrete, highly structured examples of higher-dimensional dynamical systems on K3 surfaces that semi-conjugate to plane dynamics, enriching the connections between line arrangements, matroid theory, and modular geometry.

Abstract

This paper further studies the matroid realization space of a specific deformation of the regular $n$-gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular surfaces $Ξ_{1}(n)$ over the modular curve $X_1(n)$. Here, we focus on the peculiar cases when $n=7,8$ in more detail. We obtain concrete quartic surfaces in $\mathbb{P}^3$ equipped with a dominant rational self-map stemming from an operator on line arrangements, which yields K3 surfaces with a dynamical system that is semi-conjugated to the plane.

Dynamical systems on some elliptic modular surfaces via operators on line arrangements

TL;DR

The paper investigates dynamical systems arising from operator-induced transformations on line arrangements, linking matroid realization spaces to elliptic modular surfaces. For the peculiar cases n=7 and n=8, it constructs explicit quartic models in P^3, identifies the K3 surfaces as Ξ_{1}(7) and Ξ_{1}(8), and exhibits dominant rational self-maps of degree 4 that are semi-conjugate to planar dynamics via a double-cover construction. The work combines explicit algebraic geometry with period-map techniques to realize the moduli spaces as open subschemes of modular surfaces, analyzes automorphism groups, and demonstrates the existence of periodic line-arrangement configurations. These results provide concrete, highly structured examples of higher-dimensional dynamical systems on K3 surfaces that semi-conjugate to plane dynamics, enriching the connections between line arrangements, matroid theory, and modular geometry.

Abstract

This paper further studies the matroid realization space of a specific deformation of the regular -gon with its lines of symmetry. Recently, we obtained that these particular realization spaces are birational to the elliptic modular surfaces over the modular curve . Here, we focus on the peculiar cases when in more detail. We obtain concrete quartic surfaces in equipped with a dominant rational self-map stemming from an operator on line arrangements, which yields K3 surfaces with a dynamical system that is semi-conjugated to the plane.
Paper Structure (22 sections, 16 theorems, 52 equations, 2 figures)

This paper contains 22 sections, 16 theorems, 52 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Line arrangements and the operator $\Lambda_{\mathfrak{n,m}}$
  4. Matroids and the period map of the moduli of a matroid
  5. Degree two K3 surfaces semi-conjugated to the plane
  6. The heptagon
  7. $\Lambda_{\{2\},\{3\}}$ is a rational self-map on $\mathcal{R}_{7}$ and $\mathfrak{U}_{7}$
  8. Definition of the matroid $M_{7}$
  9. Equation of the quartic surface $Z_7$ and realization space of $M_{7}$
  10. The open surface $\mathcal{R}_{7}$ inside $Z_7$
  11. The degree of $\lambda_{\{2\},\{3\}}$
  12. Action of ${\rm Aut}(M_{7})$ on the K3 surface $Z_7^{s}$
  13. Fibration preserved by $\lambda_{\{2\},\{3\}}$ and the elliptic modular surface $\Xi_{1}(7)$
  14. The K3 surface $Z_7^{s}$ is semi-conjugated to the plane
  15. The Octagon and the operator $\Lambda_{\{2\},\{3,4\}}$
  16. ...and 7 more sections

Key Result

Theorem 1

For $n\in\{7,8\}$, the dynamical system $(\mathcal{R}_{n},\lambda)$ is semi-conjugated to $(\mathbb{P}^{2},F)$ where $F:\mathbb{P}^{2}\dashrightarrow\mathbb{P}^{2}$ is an explicitly described rational self map; the dominant rational map $\pi:\mathcal{R}_{n}\to\mathbb{P}^{2}$ such that $\pi\circ\lamb

Figures (2)

  • Figure 3.1: The matroid $M_{7}$ whose construction is based on the regular heptagon.
  • Figure 4.1: Matroid $M_8$ ($\ell_i$ and $\ell_{i+4}$ meet at infinity at point $p_{i,i+4}$).

Theorems & Definitions (35)

  • Theorem 1
  • Example 2
  • Proposition 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof : Proof of \ref{['thm:action-L23-auto']}
  • Proposition 6
  • proof
  • Corollary 7
  • ...and 25 more