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Ordinary lines, point degrees, and matroid density

  • Matthew E. Kroeker
  • Ben Lund
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Abstract

For every set of n points in ℂd with full affine span, we prove that some point lies on at least n+1−2n/(d+1) spanned lines, which is best possible in every dimension. By successive contraction, this implies the sharp bound r+(ℓ−3)binom r2 on the number of elements of a simple rank-r complex-representable matroid excluding U2,ℓ as a minor, which removes the sufficiently-large-rank assumption from the bound of Geelen, Nelson, and Walsh. An arithmetic application shows that rank-r integer matrices whose maximal minors have absolute value at most Δ have at most O(Δr2) distinct columns, resolving a question of Averkov and Schymura. We prove that a set of n points in ℂ3 determines at least n2/40 ordinary lines whenever no plane contains more than 2n/3 of the points. This answers a question of Basit, Dvir, Saraf, and Wolf. We also prove upper bounds on average line length and lower bounds on the number of points incident to many spanned lines in high-dimensional point sets. The proofs combine contraction of concentrated flats with normalization of the point vectors and second- and fourth-moment estimates on their pairwise correlations. This approach builds on the design matrix method.

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Cite

BibTeXRIS
BibTeX
@misc{kroeker2026ordinary,
  title = {{Ordinary lines, point degrees, and matroid density}},
  author = {Matthew E. Kroeker and Ben Lund},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21415},
  eprint = {2610.21415},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21415},
}
RIS
TY  - UNPB
TI  - Ordinary lines, point degrees, and matroid density
AU  - Matthew E. Kroeker
AU  - Ben Lund
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21415
UR  - https://arxiv.org/abs/2610.21415
AB  - For every set of n points in ℂ^d with full affine span, we prove that some point lies on at least n+1−2n/(d+1) spanned lines, which is best possible in every dimension. By successive contraction, this implies the sharp bound r+(ℓ−3)binom r2 on the number of elements of a simple rank-r complex-representable matroid excluding U_2,ℓ as a minor, which removes the sufficiently-large-rank assumption from the bound of Geelen, Nelson, and Walsh. An arithmetic application shows that rank-r integer matrices whose maximal minors have absolute value at most Δ have at most O(Δr^2) distinct columns, resolving a question of Averkov and Schymura. We prove that a set of n points in ℂ^3 determines at least n^2/40 ordinary lines whenever no plane contains more than 2n/3 of the points. This answers a question of Basit, Dvir, Saraf, and Wolf. We also prove upper bounds on average line length and lower bounds on the number of points incident to many spanned lines in high-dimensional point sets. The proofs combine contraction of concentrated flats with normalization of the point vectors and second- and fourth-moment estimates on their pairwise correlations. This approach builds on the design matrix method.
ER  -