Van der Waerden's Problem for Cubic Surfaces
Abstract
For a smooth cubic surface S/ℚ, the absolute Galois group acts on the $27$ lines of Sℚ̅, giving a subgroup GS⊆ W(E6). We study the distribution of GS when the twenty coefficients of a defining cubic form are chosen independently from a large interval [−H,H]. We prove that, for every ε>0, the number of smooth integral cubic forms of height at most H for which GS≠ W(E6) is Oε (H19+1/21+ε). The principal obstruction is the unique subgroup W(E6)+ of index $2$. By a theorem of Saito, membership in this subgroup is equivalent to the signed discriminant of the cubic surface being a square. A recent square-value estimate of Bonolis, Pierce, and Woo then gives the exponent 19+1/21. For each of the other five conjugacy classes of maximal subgroups of W(E6), recent results on rational points in projective thin sets give the sharper bound O(H19). We also prove the lower bound Nexc(H)≫ H16 by counting cubic surfaces containing a fixed rational line, and exhibit an H14-sized exceptional family with no rational line, arising from rational Eckardt points. These examples lead us to conjecture that Nexc(H)=H16+o(1). Finally, we investigate whether Bhargava's proof of van der Waerden's conjecture for Sn can be adapted to the index-$2$ E6-problem. A double-discriminant argument gives the desired H19+ε bound whenever the product of the strongly square-dividing primes is sufficiently large. The remaining difficulty comes from weak square divisibility.
Related
- Positive Geometries from Cubic Surfaces
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- Intrinsic mirror symmetry for orbifold cubic surfaces and the A1 spherical DAHA
- Log Calabi-Yau wall crossing for moduli of cubic surfaces
- The moduli space of cubic surfaces and its orbifold fundamental group
BibTeX
@misc{odorney2026waerdens,
title = {{Van der Waerden's Problem for Cubic Surfaces}},
author = {Evan O'Dorney},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37945},
eprint = {2610.37945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37945},
}
RIS
TY - UNPB
TI - Van der Waerden's Problem for Cubic Surfaces
AU - Evan O'Dorney
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37945
UR - https://arxiv.org/abs/2610.37945
AB - For a smooth cubic surface S/ℚ, the absolute Galois group acts on the $27$ lines of S_ℚ̅, giving a subgroup G_S⊆ W(E_6). We study the distribution of G_S when the twenty coefficients of a defining cubic form are chosen independently from a large interval [−H,H]. We prove that, for every ε>0, the number of smooth integral cubic forms of height at most H for which G_S≠ W(E_6) is O_ε (H^19+1/21+ε). The principal obstruction is the unique subgroup W(E_6)^+ of index $2$. By a theorem of Saito, membership in this subgroup is equivalent to the signed discriminant of the cubic surface being a square. A recent square-value estimate of Bonolis, Pierce, and Woo then gives the exponent 19+1/21. For each of the other five conjugacy classes of maximal subgroups of W(E_6), recent results on rational points in projective thin sets give the sharper bound O(H^19). We also prove the lower bound N_exc(H)≫ H^16 by counting cubic surfaces containing a fixed rational line, and exhibit an H^14-sized exceptional family with no rational line, arising from rational Eckardt points. These examples lead us to conjecture that N_exc(H)=H^16+o(1). Finally, we investigate whether Bhargava's proof of van der Waerden's conjecture for S_n can be adapted to the index-$2$ E_6-problem. A double-discriminant argument gives the desired H^19+ε bound whenever the product of the strongly square-dividing primes is sufficiently large. The remaining difficulty comes from weak square divisibility.
ER -