On the number of ordinary lines determined by sets in complex space

Abdul Basit, Zeev Dvir, Shubhangi Saraf, Charles Wolf

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Kelly's theorem states that a set of n points affinely spanning C3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n - 1 points in a plane and one point outside the plane (in which case there are at least n - 1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.

Original languageEnglish (US)
Title of host publication33rd International Symposium on Computational Geometry, SoCG 2017
EditorsMatthew J. Katz, Boris Aronov
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages151-1515
Number of pages1365
ISBN (Electronic)9783959770385
DOIs
StatePublished - Jun 1 2017
Event33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia
Duration: Jul 4 2017Jul 7 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume77
ISSN (Print)1868-8969

Other

Other33rd International Symposium on Computational Geometry, SoCG 2017
CountryAustralia
CityBrisbane
Period7/4/177/7/17

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Combinatorial geometry
  • Designs
  • Incidences
  • Polynomial method

Cite this

Basit, A., Dvir, Z., Saraf, S., & Wolf, C. (2017). On the number of ordinary lines determined by sets in complex space. In M. J. Katz, & B. Aronov (Eds.), 33rd International Symposium on Computational Geometry, SoCG 2017 (pp. 151-1515). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 77). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2017.15