Arcs in the plane over a prime field lie mostly on a cubic curve
Let $p$ be a prime and $A \subset \mathbb{F}_p^2$ an arc: no three of its points are collinear. The largest arcs are conics, of size $p + O(1)$; Voloch, building on Segre, showed that every arc of size greater than $\frac{44}{45}p$ lies in a conic, and below $\frac12 p$ there are genuinely cubic examples. Over $\mathbb{F}_q$ with $q = p^2$ the analogous statement is false, because of arcs arising from Hermitian curves.
Problem. Must all but $o(p)$ points of $A$ lie on a single cubic curve?
Source: B. Green, Approximate algebraic structure, Proc. ICM 2014. Listed as Problem 68 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- CO — Combinatorics
- Secondary
- AG, NT
- Keywords
- algebraic curves, arcs, finite geometry, no-three-in-line
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@inproceedings{green2014icm,
author = {Green, Ben},
title = {Approximate algebraic structure},
booktitle = {Proceedings of the International Congress of Mathematicians (Seoul 2014), Vol. I},
publisher = {Kyung Moon Sa},
pages = {341--367},
year = {2014},
eprint = {1404.0093},
archivePrefix = {arXiv}
}
@article{segre1955,
author = {Segre, Beniamino},
title = {Ovals in a finite projective plane},
journal = {Canadian Journal of Mathematics},
volume = {7},
pages = {414--416},
year = {1955},
doi = {10.4153/CJM-1955-045-x}
}
@article{voloch1990,
author = {Voloch, Jos{\'e} Felipe},
title = {Arcs in projective planes over prime fields},
journal = {Journal of Geometry},
volume = {38},
number = {1-2},
pages = {198--200},
year = {1990},
doi = {10.1007/BF01222904}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 68; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00950, "Arcs in the plane over a prime field lie mostly on a cubic curve", clopen.solutions.