Highly incidental patterns on a quadratic hypersurface in $\mathbb{R}^4$

In [Sharir and Solomon 2015], Sharir and Solomon showed that the number of incidences between $m$ distinct points and $n$ distinct lines in $\mathbb R^4$ is $$O^*\left(m^{2/5}n^{4/5}+ m^{1/2}n^{1/2}q^{1/4} + m^{2/3}n^{1/3}s^{1/3} + m + n\right),$$ provided that no 2-flat contains more than $s$ lines, and no hyperplane or quadric contains more than $q$ lines, where the $O^*$ hides a multiplicative factor of $2^{c\sqrt {\log m}}$ for some absolute constant $c$. In this paper we prove that, for integers $m,n,$ satisfying $n^{9/8}<m<n^{3/2}$, there exist $m$ points and $n$ lines on the quadratic hypersurface in $\mathbb{R}^4$ $$\{(x_1,x_2,x_3,x_4)\in \mathbb R^4 \mid x_1 = x_2^2 + x_3^2 - x_4^2\},$$ such that (i) at most $s=O(1)$ lines lie on any 2-flat, (ii) at most $q=O(n/m^{1/3})$ lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is $\Theta(m^{2/3}n^{1/2})$, which is asymptotically larger than the upper bound by Sharir and Solomon. This shows that the assumption that no quadric contains more than $q$ lines (in the above mentioned theorem of Sharir and Solomon) is necessary in this regime of $m$ and $n$. By a suitable projection from this quadratic hypersurface onto $\mathbb{R}^3$, we obtain $m$ points and $n$ lines in $\mathbb{R}^3$, with at most $s=O(1)$ lines on a common plane, such that the number of incidences between the $m$ points and the $n$ lines is $\Theta(m^{2/3}n^{1/2})$. It remains an interesting question to determine if this bound is also tight in general.