  We consider quadric surface fibrations over curves, defined over
algebraically closed and finite fields. Our goal is to understand, in geometric
terms, spaces of sections for such fibrations. We analyze varieties of maximal
isotropic subspaces in the fibers as P^1-bundles over the discriminant double
cover. When the P^1-bundle is suitably stable, we deduce effective estimates
for the heights of sections over finite fields satisfying various approximation
conditions. We also discuss the behavior of the spaces of sections as the base
of the fibration acquires singularities.
