  We prove that for a generic element in a nonhyperelliptic component of an
abelian stratum $\mathcal{H}_g(\mu)$ in genus $g$, the underlying curve has
maximal gonality. We extend this result to the case of quadratic strata when
the partition $\mu$ has positive entries. As a consequence we deduce that all
nonhyperelliptic components of $\mathcal{H}_9(\mu)$ are uniruled when $\mu$ is
a positive partition of 16 and all nonhyperelliptic components of
$\mathcal{H}^2_g(\mu)$ are uniruled when $\mu$ is a positive partition of
$4g-4$ and either $3\leq g\leq5$ or $g=6$ and $l(\mu)\geq 4$.
