Abstract
We prove an incidence theorem for points and planes in the projective space P3 over any field F, whose characteristic p ≠ 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 p > 0.
This yields a bound on the number of incidences between m points and n planes in P3, with m ≥ n as O (m√n + 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 S ⊆ 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 S ⊆ 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.
This yields a bound on the number of incidences between m points and n planes in P3, with m ≥ n as O (m√n + 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 S ⊆ 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 S ⊆ 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 language | English |
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Pages (from-to) | 219–254 |
Number of pages | 36 |
Journal | Combinatorica |
Volume | 38 |
Issue number | 1 |
Early online date | 10 Jan 2017 |
DOIs | |
Publication status | Published - Feb 2018 |
Bibliographical note
Accepted author manuscript a 25pp. revised version: a remark, a lemma, and a theorem have been added.Keywords
- math.CO
- 68R05
- 11B75
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Dr Misha Rudnev
- School of Mathematics - Associate Professor in Mathematics
- Number theory and combinatorics
- Pure Mathematics
Person: Academic , Member