The Reye geometry inside the 64 lines of the Schur quartic
Abstract
We identify the classical geometry hidden in the Naskręcki--Pokora (244,323) configuration on the Schur quartic. Using Höhn's identification of the $24$ selected lines with the $24$ roots of D4, the antipodal involution on the roots induces a fixed-point-free quotient of the incidence configuration, and this quotient is precisely the classical Reye configuration. We also determine the symmetry of the complete $64$-line incidence geometry: its automorphism group has order $4608$, the two Naskręcki--Pokora configurations form a single orbit, and the stabilizer of either has order $2304$ (projectively, $576$). The $64$ lines extend canonically to a $176$-line arrangement carried by six projectively equivalent Schur quartics, with 176=16+16+9·16 and induced surface permutation group S3× S3. Finally, the antipodal quotient itself extends coherently through this six-quartic geometry: on each Schur quartic it produces two Reye configurations sharing the same $16$-element incidence skeleton, and on the full $176$-line arrangement it gives a compatible global quotient. This reveals a precise incidence-theoretic connection with classical desmic geometry, while showing that this connection is not a literal identification with the two Reye configurations arising from the classical desmic construction in P3.
Related
- A tree bijection for Eulerian orientations on quartic maps
- The Frohmanian Symplectic Tether: Canonical Uniqueness on Coadjoint Orbits of SDiff(T3), Non-Degenerate Casimir Coupling, and Conditional Global Regularity via Quartic Young Absorption in Lean 4
- Large values of cubic and quartic Dirichlet L-functions of prime moduli
- Very general quartic sixfolds are not stably rational
- Universal torsors over quartic del Pezzo surfaces and stable rationality
BibTeX
@misc{nurowski2026reye,
title = {{The Reye geometry inside the 64 lines of the Schur quartic}},
author = {Paweł Nurowski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10751},
eprint = {2609.10751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10751},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Reye geometry inside the 64 lines of the Schur quartic
AU - Paweł Nurowski
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.10751
UR - https://arxiv.org/abs/2609.10751
AB - We identify the classical geometry hidden in the Naskręcki--Pokora (24_4,32_3) configuration on the Schur quartic. Using Höhn's identification of the $24$ selected lines with the $24$ roots of D_4, the antipodal involution on the roots induces a fixed-point-free quotient of the incidence configuration, and this quotient is precisely the classical Reye configuration. We also determine the symmetry of the complete $64$-line incidence geometry: its automorphism group has order $4608$, the two Naskręcki--Pokora configurations form a single orbit, and the stabilizer of either has order $2304$ (projectively, $576$). The $64$ lines extend canonically to a $176$-line arrangement carried by six projectively equivalent Schur quartics, with 176=16+16+9·16 and induced surface permutation group S_3× S_3. Finally, the antipodal quotient itself extends coherently through this six-quartic geometry: on each Schur quartic it produces two Reye configurations sharing the same $16$-element incidence skeleton, and on the full $176$-line arrangement it gives a compatible global quotient. This reveals a precise incidence-theoretic connection with classical desmic geometry, while showing that this connection is not a literal identification with the two Reye configurations arising from the classical desmic construction in P^3.
ER -