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. In Höhn's $D_4$ labelling, the antipodal involution on the $24$ 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, $576$). Finally, 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 $S_3\times S_3$.