Incidences between points and lines on two- and three-dimensional varieties

Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${\mathbb{R}}^4$, such that the points of $P$ lie on an algebraic three-dimensional surface of degree $D$ that does not contain hyperplane or quadric components, and no 2-flat contains more than $s$ lines of $L$. We show that the number of incidences between $P$ and $L$ is $$I(P,L)=O(m^{1\mathrm{/}2}{n}^{1\mathrm{/}2}D+m^{2\mathrm{/}3}{n}^{1\mathrm{/}3}{s}^{1\mathrm{/}3}+nD+m),$$ for some absolute constant of proportionality. This significantly improves the bound of the authors, for arbitrary sets of points and lines in ${\mathbb{R}}^4$, when $D$ is not too large. The same bound holds when the three-dimensional surface is embedded in any higher dimensional space. For the proof of this bound, we revisit certain parts of [Sharir-Solomon16], combined with the following new incidence bound. Let $P$ be a set of $m$ points and $L$ a set of $n$ lines in ${\mathbb{R}}^d$, for $d\ge 3$,whichlieinacommontwo-dimensionalalgebraicsurfaceofdegree$D$(assumedtobe$\ll n^{1/2}$) that does not contain any 2-flat, so that no 2-flat contains more than $s$ lines of $L$ (here we require that the lines of $L$ also be contained in the surface). 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}D^{2/3}s^{1/3}

• m + n\right). $$When$d=3$, this improves the bound of Guth and Katz for this special case, when $D\ll n^{1\mathrm{/}2}$. Moreover, the bound does not involve the term $O(nD)$, that arises in most standard approaches, and its removal is a significant aspect of our result. Finally, we also obtain (slightly weaker) variants of both results over the complex field. For two-dimensional varieties, the bound is as in the real case, with an added term of $O(D^3)$. For three-dimensional varieties, the bound is as in the real case, with an added term of $O(D^6)$.