2014/07/01 by Misha Rudnev, Rudnev, Misha
Mathematics · #11B75 #68R05 #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1407.0426
openalex publication_date 2014/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove an incidence theorem for points and planes in the projective space\n mathbb P3 over any field mathbb F, whose characteristic p\≠ 2. An\nincidence is viewed as an intersection along a line of a pair of two-planes\nfrom two canonical rulings of the Klein quadric. The Klein quadric can be\ntraversed by a generic hyperplane, yielding a line-line incidence problem in a\nthree-quadric, the Klein image of a regular line complex. This hyperplane can\nbe chosen so that at most two lines meet. Hence, one can apply an algebraic\ntheorem of Guth and Katz, with a constraint involving p if p>0.\n This yields a bound on the number of incidences between m points and n\nplanes in mathbb P3, with m\≥ n as O
left(m
sqrtn+ m k
right),\nwhere k is the maximum number of collinear planes, provided that n=O(p2)\nif p>0. Examples show that this bound cannot be improved without additional\nassumptions.\n This gives one a vehicle to establish geometric incidence estimates when\np>0. For a non-collinear point set S\⊆ mathbb F2 and a\nnon-degenerate symmetric or skew-symmetric bilinear form \ω, the number\nof distinct values of \ω on pairs of points of S is\n\Ω\[\min\(|S|\(2)/(3),p\)\]. This is also the\nbest known bound over mathbb R, where it follows from the\nSzemer 'edi-Trotter theorem. Also, a set S\⊆ mathbb F3, not\nsupported in a single semi-isotropic plane contains a point, from which\n\Ω\[\min\(|S|\(1)/(2),p\)\] distinct distances to\nother points of S are attained.\n