Abstract: Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Pl\"ucker space of lines in $R^3$, 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 $O(n^{4/3+\epsilon}/r^2)$, for $r \ge 3$ and for any $\epsilon>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at mos...
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