  On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a
symmetric relatively ample line bundle defines a fiberwise N\'eon-Tate height.
If the base curve is inside a projective space, we also have a height on its
$\overline{\mathbb Q}$-points that serves as a measure of each fiber, an
abelian variety. Silverman proved an asymptotic equality between these two
heights on a curve in the abelian scheme. In this paper we prove an inequality
between these heights on a subvariety of any dimension of the abelian scheme.
As an application we prove the Geometric Bogomolov Conjecture for the function
field of a curve defined over $\overline{\mathbb Q}$. Using Moriwaki's height
we sketch how to extend our result when the base field of the curve has
characteristic 0.
