Szemerédi--Trotter-type theorems in dimension 3

2014/05/06 by János Kollár, Kollár, János · 1 citation
Mathematics · #05A20 #14N20 #51E20 #52B55 #52C10 #68R05 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO #msc:05A20 #msc:14N20 #msc:51E20 #msc:52B55 #msc:52C10 #msc:68R05

paper · pdf · doi:10.48550/arxiv.1405.2243

This paper supersedes arXiv:1404.4613. Version2: references updated and small changes made. arXiv admin note: substantial text overlap with arXiv:1404.4613 Version 3: Many changes and a new section added on ruled surfaces

arxiv created 2014/12/03 · arxiv updated 2014/12/05

Abstract

We estimate the number of incidences in a configuration of m lines and n points in dimension 3. The main term is mn^1/3 if we work over the real or complex numbers but mn^2/5 over finite fields. Both of these are optimal, aside from a multiplicative constant that is at most 5.

Citations

• Incidences of lines in P³ and the arithmetic genus of curves

Cited by

• The Entropic Sum-Product Phenomenon

Related

• The Szemerédi-Trotter theorem over arbitrary field of characteristic zero
• Multi-parameter Szemerédi-Trotter-type theorems and applications in finite fields
• On forbidden configurations in point-line incidence graphs
• Generalizations of the Szemerédi-Trotter Theorem
• Incidences between points and lines in three dimensions