On the number of incidences between points and planes in three dimensions

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We prove an incidence theorem for points and planes in the projective space P3 over any field F, whose characteristic ≠ 2 An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if > 0. 
This yields a bound on the number of incidences between m points and n planes in P3, with  n as (m√+ mk),where k is the maximum number of collinear planes, provided that n=O(p2) if p>0. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when p>0. For a non-collinear point set F2 and a non-degenerate symmetric or skew-symmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is Ω[min(|S|2/3,p)]. This is also the best known bound over R, where it follows from the Szemerédi-Trotter theorem. Also, a set F3, not supported in a single semi-isotropic plane contains a point, from which Ω[min(|S|½,p)] distinct distances to other points of S are attained.
Original languageEnglish
Pages (from-to)219–254
Number of pages36
Issue number1
Early online date10 Jan 2017
Publication statusPublished - Feb 2018

Bibliographical note

Accepted author manuscript a 25pp. revised version: a remark, a lemma, and a theorem have been added.


  • math.CO
  • 68R05
  • 11B75


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