  The Bounded Height Conjecture of Bombieri, Masser, and Zannier states that
for any sufficiently generic algebraic subvariety of a semiabelian
$\overline{\mathbb{Q}}$-variety $G$ there is an upper bound on the Weil height
of the points contained in its intersection with the union of all algebraic
subgroups having (at most) complementary dimension in $G$. This conjecture has
been shown by Habegger in the case where $G$ is either a multiplicative torus
or an abelian variety. However, there are new obstructions to his approach if
$G$ is a general semiabelian variety. In particular, the lack of Poincar\'e
reducibility means that quotients of a given semiabelian variety are intricate
to describe. To overcome this, we study directly certain families of line
bundles on $G$. This allows us to demonstrate the conjecture for general
semiabelian varieties.
