Hostname: page-component-6bf8c574d5-t27h7 Total loading time: 0 Render date: 2025-02-26T10:59:36.588Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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
Get access Rights & Permissions [Opens in a new window]

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.)
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 19
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bosma, W., Cannon, J. and Playoust, C., The magma algebra system. I. The user language , J. Symbolic Comput. 24 (1997), no. 3–4, 235–265.CrossRefGoogle Scholar
Cantat, S. and Dolgachev, I., Rational surfaces with a large group of automorphisms , J. Amer. Math. Soc. 25 (2012), no. 3, 863–905.CrossRefGoogle Scholar
Corey, D., Kühne, L. and Schröter, B., Matroids in OSCAR, preprint, arXiv:2311.08792.Google Scholar
Huybrechts, D., Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, Vol. 158, Cambridge University Press, Cambridge, 2016, xi+485 pp.CrossRefGoogle Scholar
Kühne, L. and Roulleau, X., Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids, preprint, arXiv:2312.03470.Google Scholar
Lecacheux, O., Weierstrass equations for all elliptic fibrations on the modular K3 surface associated to ${\varGamma}_1(7)$ , Rocky Mountain J. Math. 45 (2015), no. 5, 1481–1509.CrossRefGoogle Scholar
Naruki, I., On a K3 surface which is a ball quotient, Max Planck Institute for Mathematics Preprint Series, 1985. https://archive.mpimbonn.mpg.de/id/eprint/2113/ Google Scholar
Oxley, J., Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, Vol. 21, 2011, xiv+684 pp.CrossRefGoogle Scholar
Roulleau, X., On some operators acting on line arrangements and their dynamics, preprint, arXiv:2306.01053.Google Scholar
Schütt, M., K3 surfaces with Picard rank 20 , Algebra Number Theory. 4 (2010), no. 3, 335–356.CrossRefGoogle Scholar
Shioda, T., Elliptic modular surfaces , J. Math. Soc. Japan. 24 (1972), 20–59.CrossRefGoogle Scholar
Suciu, A. I., Hyperplane arrangements and Milnor fibrations , Ann. Fac. Sci Toulouse Math.(6) 23 (2014), no. 2, 417–481.CrossRefGoogle Scholar
Top, J. and Yui, N., Explicit equations of some elliptic modular surfaces , Rocky Mountain J. Math. 37 (2007), no. 2, 663–687.CrossRefGoogle Scholar
Ujikawa, M., The automorphism group of the singular K3 surface of discriminant 7 , Comment. Math. Univ. St. Pauli. 62 (2013), no. 1, 11–29.Google Scholar
Voisin, C., Intrinsic pseudo-volume forms and K-correspondences, The Fano Conference, Università di Torino, Turin, 2004, 761–792.Google Scholar
Yoshinaga, M., Freeness of hyperplane arrangements and related topics , Ann. Fac. Sci. Toulouse Math. (6). 23 (2014), no. 2, 483–512.CrossRefGoogle Scholar