@inproceedings{c90e688778c34cfeb96019eaae0a6d0c,

title = "On the number of ordinary lines determined by sets in complex space",

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.",

keywords = "Combinatorial geometry, Designs, Incidences, Polynomial method",

author = "Abdul Basit and Zeev Dvir and Shubhangi Saraf and Charles Wolf",

note = "Funding Information: ∗ A full version of the paper is available at http://arxiv.org/abs/1611.08740. † Work of the second author was supported by NSF CAREER award DMS-1451191 and NSF grant CCF-1523816. Work by the third and fourth authors was supported in part by NSF grant CCF-1350572. Publisher Copyright: {\textcopyright} Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf.; 33rd International Symposium on Computational Geometry, SoCG 2017 ; Conference date: 04-07-2017 Through 07-07-2017",

year = "2017",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SoCG.2017.15",

language = "English (US)",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

pages = "151--1515",

editor = "Katz, {Matthew J.} and Boris Aronov",

booktitle = "33rd International Symposium on Computational Geometry, SoCG 2017",

address = "Germany",

}