arXiv:2610.14099 · v3

Symmetric, rational and infinitely many $(23_4)$ configurations, and the end of the existence problem for geometric $(n_4)$ configurations

首次提交 2026-10-09 03:59

版本更新 2026-11-08 02:01

雷达观测 2026-11-10 11:14

摘要

A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, and successive work of Bokowski, Grünbaum, Schewe, Pilaud and Cuntz settled every case but $n=23$. We settle this last case by bounding the symmetry of such configurations and classifying the most symmetric ones exactly. We prove that for every prime $n\equiv3\pmod4$ the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, explaining why symmetric constructions failed at $n=23$, and the same argument bounds the symmetry of $(n_5)$ configurations for primes $n\not\equiv1\pmod5$. Reducing the Klein four-group case to 4021 bilinear polynomial systems and solving all of them exactly with Gröbner bases, we show that exactly two $(23_4)$ configurations reach this bound up to projective equivalence and that there are no others even over $\mathbb{C}$. Both are self-polar; the first has integer coordinates and the second is defined over $\mathbb{Q}(\sqrt{17})$ but not over $\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\ge18$ and $n\ne19$. The same classification recovers Cuntz's $(22_4)$ and $(26_4)$ configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$. Among self-polar $(23_4)$ configurations with a single involution we find $50$ more, defined over number fields of degree at most $7$. Four of them deform in one-parameter families, and computing three of these exactly, one birational to the modular curve $X_1(11)$, we prove that there are infinitely many geometric $(23_4)$ configurations, among them infinitely many with rational coordinates that are self-polar and have no projective symmetry.

作者备注：To be presented at the 2027 Joint Mathematics Meetings, Chicago (AMS Contributed Paper Session on Combinatorics, I, January 12, 2027)