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

Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties