Incidences between points and lines on a two-dimensional variety

Abstract

We present a direct and fairly simple proof of the following incidence bound: Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${\mathbb{R}}^d$, for $d\ge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does not contain any 2-flat, so that no 2-flat contains more than $s\le D$ lines of $L$. Then the number of incidences between $P$ and $L$ is $$I(P,L)=O\left(m^{1/2}{n}^{1/2}{D}^{1/2}+m^{2/3}\min \{n,D^2{\}}^{1/3}{s}^{1/3}+m+n\right).$$ When $d=3$, this improves the bound of Guth and Katz~\cite{GK2} for this special case, when $D$ is not too large. A supplementary feature of this work is a review, with detailed proofs, of several basic (and folklore) properties of ruled surfaces in three dimensions.

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