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Is there any characterization of abelian varieties appearing as direct factor of the Jacobian of some curve?

Are there some special kind of abelian varieties that are known to be direct factor of the Jacobian of some curve?

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    $\begingroup$ Any abelian variety $A$ is isogenous to a direct factor of a Jacobian: just take a sufficiently ample curve $C\subset A$, so that $A$ embeds into $JC$, and apply Poincare reducibility theorem. I doubt that you can characterize those for which the isogeny is actually an isomorphism. $\endgroup$
    – abx
    Commented Jul 7, 2017 at 12:36
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    $\begingroup$ @abx: Of course, to make the argument work over finite fields, one should use Poonen's refinement of Bertini theorems to work over finite fields (this being one of the applications in his original paper). $\endgroup$
    – nfdc23
    Commented Jul 7, 2017 at 13:39

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