  We improve the lower bound on the $d$-dimensional rectilinear crossing number
of the complete $d$-uniform hypergraph having $2d$ vertices to
$\Omega\left(\dfrac{(4\sqrt{2}/3^{3/4})^d}{d}\right)$ from $\Omega(2^d
\sqrt{d})$. We also establish that the $3$-dimensional rectilinear crossing
number of a complete $3$-uniform hypergraph having $n \geq 9$ vertices is at
least $\dfrac{43}{42}\dbinom{n}{6}$.
  We prove that the maximum number of crossing pairs of hyperedges in a
$4$-dimensional rectilinear drawing of the complete $4$-uniform hypergraph
having $n$ vertices is $13\dbinom{n}{8}$. We also prove that among all
$4$-dimensional rectilinear drawings of a complete $4$-uniform hypergraph
having $n$ vertices, the number of crossing pairs of hyperedges is maximized if
all its vertices are placed at the vertices of a $4$-dimensional neighborly
polytope. Our result proves the conjecture by Anshu et al. [Anshu,
Gangopadhyay, Shannigrahi, and Vusirikala, 2017] for $d=4$.
  We prove that the maximum $d$-dimensional rectilinear crossing number of a
complete $d$-partite $d$-uniform balanced hypergraph is
$(2^{d-1}-1){\dbinom{n}{2}}^d$. We then prove that finding the maximum
$d$-dimensional rectilinear crossing number of an arbitrary $d$-uniform
hypergraph is NP-hard.
  We give a randomized scheme to create a $d$-dimensional rectilinear drawing
of a $d$-uniform hypergraph $H$ such that, in expectation the total number of
crossing pairs of hyperedges is a constant fraction of the maximum
$d$-dimensional rectilinear crossing number of $H$.
