  We show that any polarized abelian variety over a finite field is covered by
a Jacobian whose dimension is bounded by an explicit constant. We do this by
first proving an effective version of Poonen's Bertini theorem over finite
fields, which allows us to show the existence of smooth curves arising as
hypersurface sections of bounded degree and genus. Additionally, we show that
for simple abelian varieties a better bound is possible. As an application of
these results we show that if $E$ is an elliptic curve over a finite field then
for any $n\in \mathbb{N}$ there exist smooth curves of bounded genus whose
Jacobians have a factor isogenous to $E^n$.
