  The number $Z(n):=\lfloor n/2\rfloor\lfloor (n-1)/2\rfloor$ is the smallest
number of crossings in a simple planar drawing of $K_{2,n}$ in which both
vertices on the 2-side have the same clockwise rotation. For two vertices $u,v$
on the $q$-side of a simple drawing of $K_{p,q}$, let
$\operatorname{cr}_D(u,v)$ denote the total number of crossings that edges
incident with $u$ have with edges incident with $v$.
  We show that in any simple drawing $D$ of $K_{p,q}$ in a surface $\Sigma$ the
number of pairs of vertices on the $q$-side of $K_{p,q}$ having
$\operatorname{cr}_D(u,v)<Z(p)$ is bounded as a function of $p$ and $\Sigma$.
As a consequence, we also show that, for a fixed integer $p$ and surface
$\Sigma$, there exists a finite set of drawings $\mathcal{D}(p,\Sigma)$ of
complete bipartite graphs such that, for each $q$, a crossing-minimal drawing
of $K_{p,q}$ can be obtained by "duplicating vertices" in some drawing from
$\mathcal D(p,\Sigma)$.
