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 $dge 3$, which lie in a common algebraic two-dimensional surface of degree $D$ that does no
t 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 $$ I(P,L)=Oleft(m^{1/2}n^{1/2}D^{1/2} + m^{2/3}min{n,D^{2}}^{1/3}s^{1/3} + m + nright). $$ 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.
We give a fairly elementary and simple proof that shows that the number of incidences between $m$ points and $n$ lines in ${mathbb R}^3$, so that no plane contains more than $s$ lines, is $$ Oleft(m^{1/2}n^{3/4}+ m^{2/3}n^{1/3}s^{1/3} + m + nright) $
$ (in the precise statement, the constant of proportionality of the first and third terms depends, in a rather weak manner, on the relation between $m$ and $n$). This bound, originally obtained by Guth and Katz~cite{GK2} as a major step in their solution of Erd{H o}ss distinct distances problem, is also a major new result in incidence geometry, an area that has picked up considerable momentum in the past six years. Its original proof uses fairly involved machinery from algebraic and differential geometry, so it is highly desirable to simplify the proof, in the interest of better understanding the geometric structure of the problem, and providing new tools for tackling similar problems. This has recently been undertaken by Guth~cite{Gu14}. The present paper presents a different and simpler derivation, with better bounds than those in cite{Gu14}, and without the restrictive assumptions made there. Our result has a potential for applications to other incidence problems in higher dimensions.
