On rich points and incidences with restricted sets of lines in 3-space

Abstract: Let $L$ be a set of $n$ lines in $R<sup>3$ that is contained, when represented as points in the four-dimensional Pl\"ucker space of lines in $R<sup>3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n<sup>{4/3+\epsilon}/r<sup>2)$, for $r \ge 3$ and for any $\epsilon&gt;0$, and, if at most $n<sup>{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n<sup>{4/3+\epsilon})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$. As an application, we deduce (with an $\epsilon$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R<sup>3$ is $O(m+n)$. \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R<sup>3$ is $$O\left(m^{{3/5}n^{3/5}} + (m^{{11/15}n^{2/5}} + m^{{1/3}n^{{2/3})s^{1/3}}} + m + n \right)$$, provided that no plane contains more than $s$ of the points. When $$s = O(\min{n^{{3/5}/m^{2/5},} m<sup>{1/2}})$$, the bound becomes $O(m<sup>{3/5}n<sup>{3/5}+m+n)$. As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R<sup>4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{{3/5}n^{3/5}} + m + n)$. The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.