  We introduce the notion of pseudo-N\'eron model and give new examples of
varieties admitting pseudo-N\'eron models other than Abelian varieties. As an
application of pseudo-N\'eron models, given a scheme admitting a finite
morphism to an Abelian scheme over a positive-dimensional base, we prove that
for a very general genus-0, degree-d curve in the base with d sufficiently
large, every section of the scheme over the curve is contained in a unique
section over the entire base.
