Schur四次曲线64条直线内部的Reye几何
The Reye geometry inside the 64 lines of the Schur quartic
- Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文识别出Schur四次曲线上Naskręcki--Pokora构型隐藏的经典Reye几何，确定64线关联几何的自同构群阶为4608，并典范扩展至176线排列。
AI中文摘要：
我们识别出Naskręcki--Pokora $(24_4,32_3)$构型在Schur四次曲线上隐藏的经典几何。在Höhn的$D_4$标记下，24个根上的对径对合诱导了关联构型的无不动点商，该商恰好是经典的Reye构型。我们还确定了完整64条直线关联几何的对称性：其自同构群阶为4608，两个Naskręcki--Pokora构型形成单一轨道，且任一构型的稳定子阶为2304（射影意义下为576）。最后，这64条直线典范地扩展为由六个射影等价的Schur四次曲线承载的176条直线排列，其中$176=16+16+9\cdot16$，诱导曲面置换群为$S_3\times S_3$。
英文摘要：
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 $P^3$.
