$\mathbb{P}^{2n + 1}_K$中斜$n$平面配置产生的共线完全集和有限子群
Collinearly complete sets and finite subgroups from configurations of skew $n$-planes in $\mathbb P^{2n+1}_K$
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中文总结 AI 辅助
研究$\mathbb{P}^{2n + 1}_K$中斜$n$平面配置产生的共线完全集和有限子群，通过关联广群研究其轨道几何，在特征零时有刚性结果，还证明了轨道相关性质及正特征下的不同情况。
中文摘要 AI 辅助
我们研究了由$\mathbb{P}^{2n + 1}_K$中两两斜交的$n$平面配置产生的共线完全有限点集。对于这样的配置，我们关联一个由$n$平面之间的自然共线性对应生成的广群，并研究其有限轨道几何。在特征为零的情况下，我们证明了相关群为有限循环群时的刚性结果。经过适当归一化，定义配置的矩阵可同时对角化并具有共同的两块分解。因此，一般点的轨道与每个$n$平面相交于共线集，且整个轨道包含在一个特殊的射影$3$空间中。这将此类轨道几何简化为$\mathbb{P}^3_{\mathbb{C}}$中斜线的经典情况：在特殊的$\mathbb{P}^3_{\mathbb{C}}$内，轨道是geproci的。我们还证明了一般轨道的有限并集在集合论上由一个可约曲面从$n$平面的并集中切出。最后，我们表明这种特征零的刚性在正特征下不成立。在特征为$2$时，我们构造了轨道切片为法诺平面配置的循环示例，并在射影$5$空间中展示了一个真正的高维有限非循环示例，其相关群为${\rm PGL}_3(\mathbb{F}_2)\cong {\rm PSL}_2(\mathbb{F}_7)$。
英文摘要
We study collinearly complete finite sets of points arising from configurations of pairwise skew $n$-planes in $\mathbb {P}^{2n+1}_K$. To such a configuration we associate a groupoid generated by the natural collinearity correspondences between the $n$-planes, and we investigate the geometry of its finite orbits. In characteristic zero, we prove a rigidity result for the case in which the associated group is finite cyclic. After a suitable normalization, the matrices defining the configuration are simultaneously diagonalizable and admit a common two-block decomposition. Consequently, the orbit of a general point meets each $n$-plane in a collinear set, and the full orbit is contained in a distinguished projective $3$-space. This reduces the geometry of such orbits to the classical case of skew lines in $\mathbb {P}^3_{\mathbb C}$: inside the distinguished $\mathbb{P}^3_{\mathbb C}$, the orbit is geproci. We also prove that finite unions of general orbits are cut out set-theoretically from the union of the $n$-planes by a reducible surface. Finally, we show that this characteristic-zero rigidity fails in positive characteristic. In characteristic $2$, we construct cyclic examples whose orbit slices are Fano plane configurations, and we exhibit a genuinely higher-dimensional finite non-cyclic example in projective $5$-space with associated group ${\rm PGL}_3(\mathbb F_2)\cong {\rm PSL}_2(\mathbb F_7)$.
