Symmetric, rational and infinitely many (234) configurations, and the end of the existence problem for geometric (n4) configurations
Abstract
A geometric (n4) 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≡3pmod4 the group of projective symmetries of a geometric (n4) configuration has order at most four, explaining why symmetric constructions failed at n=23, and the same argument bounds the symmetry of (n5) configurations for primes nnot≡1pmod5. 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 (234) configurations reach this bound up to projective equivalence and that there are no others even over ℂ. Both are self-polar; the first has integer coordinates and the second is defined over ℚ(√(17)) but not over ℚ. Consequently geometric (n4) configurations exist if and only if n≥18 and n≠19. The same classification recovers Cuntz's (224) and (264) configurations and finds three new combinatorial types at n=22 and $94$ at n=26. Among self-polar (234) 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 X1(11), we prove that there are infinitely many geometric (234) configurations, among them infinitely many with rational coordinates that are self-polar and have no projective symmetry.
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BibTeX
@misc{lu2026symmetric,
title = {{Symmetric, rational and infinitely many $(23_4)$ configurations, and the end of the existence problem for geometric $(n_4)$ configurations}},
author = {Wenhao Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14099},
eprint = {2610.14099},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14099},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Symmetric, rational and infinitely many (23_4) configurations, and the end of the existence problem for geometric (n_4) configurations
AU - Wenhao Lu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.14099
UR - https://arxiv.org/abs/2610.14099
AB - 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≡3pmod4 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 nnot≡1pmod5. 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 ℂ. Both are self-polar; the first has integer coordinates and the second is defined over ℚ(√(17)) but not over ℚ. Consequently geometric (n_4) configurations exist if and only if n≥18 and n≠19. 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.
ER -