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DYNAMICAL SYSTEMS ON SOME ELLIPTIC MODULAR SURFACES VIA OPERATORS ON LINE ARRANGEMENTS

Part of: Projective and enumerative geometry Curves Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  20 January 2025

LUKAS KÜHNE Open the ORCID record for LUKAS KÜHNE [Opens in a new window]  and
XAVIER ROULLEAU Open the ORCID record for XAVIER ROULLEAU [Opens in a new window]
LUKAS KÜHNE*
Affiliation:
Fakultät für Mathematik Universität Bielefeld 33615, Bielefeld Germany
XAVIER ROULLEAU
Affiliation:
CNRS, LAREMA, SFR Math-STIC Université d’Angers F-49000 Angers France xavier.roulleau@univ-angers.fr
*
lkuehne@math.uni-bielefeld.de
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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 $\Xi _{1}(n)$ 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.

Keywords

matroid realization spaceelliptic modular surfaceK3 surfacedynamical systems

MSC classification

Primary: 14N20: Configurations and arrangements of linear subspaces
Secondary: 14H50: Plane and space curves 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Information

Type
Article
Information
Nagoya Mathematical Journal , Volume 258 , June 2025 , pp. 357 - 375
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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