PreprintCC BY 4.0arXiv
Covering plane curves over finite fields with lines
Abstract
We prove that, for every fixed dβ₯ 2 and Ξ΅>0, the π½q-points of a geometrically irreducible reflexive plane curve of degree d over π½q can be covered by at most (1/d+Ξ΅)q many π½q-lines, provided that q is sufficiently large. The same conclusion holds for π½q-irreducible curves that are not geometrically irreducible, since they have only O(d2) π½q-points. In particular, for sufficiently large q, the bound applies to every π½q-irreducible plane curve of degree d when char(π½q)>d. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger--Spencer theorem on covers of nearly regular hypergraphs.
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BibTeX
@misc{asgarli2026covering,
title = {{Covering plane curves over finite fields with lines}},
author = {Shamil Asgarli and Sarthak Dassarma and Dragos Ghioca and Simon MacLean and Evan Stokdyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13042},
eprint = {2610.13042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13042},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Covering plane curves over finite fields with lines
AU - Shamil Asgarli
AU - Sarthak Dassarma
AU - Dragos Ghioca
AU - Simon MacLean
AU - Evan Stokdyk
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13042
UR - https://arxiv.org/abs/2610.13042
AB - We prove that, for every fixed dβ₯ 2 and Ξ΅>0, the π½_q-points of a geometrically irreducible reflexive plane curve of degree d over π½_q can be covered by at most (1/d+Ξ΅)q many π½_q-lines, provided that q is sufficiently large. The same conclusion holds for π½_q-irreducible curves that are not geometrically irreducible, since they have only O(d^2) π½_q-points. In particular, for sufficiently large q, the bound applies to every π½_q-irreducible plane curve of degree d when char(π½_q)>d. Our bound is asymptotically sharp. The proof combines Entin's Chebotarev density theorem for varieties over finite fields with the Pippenger--Spencer theorem on covers of nearly regular hypergraphs.
ER -