The Reye geometry inside the 64 lines of the Schur quartic

Abstract

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 $176=16+16+9\cdot16$ and induced surface permutation group $S_3\times S_3$. 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 $P^3$.