Abstract
We show that if a finite non-collinear set of points in \(\mathbb {C}^2\) lies on a family of m concurrent lines, and if one of those lines contains more than \(m-2\) points, there exists a line passing through exactly two points of the set. The bound \(m-2\) in our result is optimal. Our main theorem resolves a conjecture of Frank de Zeeuw, and generalizes a result of Kelly and Nwankpa.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berger, M.: Geometry Revealed. Springer, Heidelberg (2010)
Bokowski, J., Pokora, P.: On the Sylvester–Gallai and the orchard problem for pseudoline arrangements. Period. Math. Hungar. 77(2), 164–174 (2018)
Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)
Green, B., Tao, T.: On sets defining few ordinary lines. Discrete Comput. Geom. 50(2), 409–468 (2013)
Hirzebruch, F.: Arrangements of lines and algebraic surfaces. In: Arithmetic and Geometry, vol. 2. Progress in Mathematics, vol. 36, pp. 113–140. Birkhäuser, Boston (1983)
Kelly, L.M.: A resolution of the Sylvester–Gallai problem of J.-P. Serre. Discrete Comput. Geom. 1(2), 101–104 (1986)
Kelly, L.M., Nwankpa, S.: Affine embeddings of Sylvester–Gallai designs. J. Comb. Theory Ser. A 14, 422–438 (1973)
Langer, A.: Logarithmic orbifold Euler numbers of surfaces with applications. Proc. Lond. Math. Soc. 86(2), 358–396 (2003)
Limbos, M.: Projective embeddings of small “Steiner triple systems”. Ann. Discrete Math. 7, 151–173 (1980)
Marsden, J.E., Tromba, A.: Vector Calculus. W.H. Freeman and Company, New York (2011)
Motzkin, T.: The lines and planes connecting the points of a finite set. Trans. Am. Math. Soc. 70, 451–464 (1951)
Pokora, P.: Hirzebruch-type inequalities viewed as tools in combinatorics (2020). arXiv:1808.09167
Serre, J.-P.: Problem # 5359. Am. Math. Mon. 73, 89 (1966)
de Zeeuw, F.: Ordinary lines in space (2018). arXiv:1803.09524
Acknowledgements
Many thanks to Frank de Zeeuw for suggesting the problem and for helpful discussions along the way. Thanks to the Baruch Combinatorics REU and organizer Adam Scheffer for supporting this work and providing research mentorship. Thanks to Wilhelm Schlag for providing suggestions with regard to Lemma 2.5. The author would also like to thank an anonymous reviewer for extremely helpful and detailed comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was done as part of the 2019 CUNY Combinatorics REU, supported by NSF awards DMS-1802059 and DMS-1851420.
Rights and permissions
About this article
Cite this article
Cohen, A. A Sylvester–Gallai Result for Concurrent Lines in the Complex Plane. Discrete Comput Geom 68, 172–187 (2022). https://doi.org/10.1007/s00454-020-00256-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-020-00256-2