Abelian varieties are not quotients of low-dimension Jacobians

We prove that for any two integers $g\ge 4$ and $g^{\mathrm{\prime }}\le 2g-1$, there exist abelian varieties over $\bar{\mathbb{Q}}$ which are not quotients of a Jacobian of dimension $g^{\mathrm{\prime }}$. Our method in fact proves that most Abelian varieties satisfy this property, when counting by height relative to a fixed finite map to projective space.