  Laver, and Woodin independently, showed that models of ${\rm ZFC}$ are
uniformly definable in their set-forcing extensions, using a ground model
parameter. We investigate ground model definability for models of fragments of
${\rm ZFC}$, particularly of ${\rm ZF}+{\rm DC}_\delta$ and of ${\rm ZFC}^-$,
and we obtain both positive and negative results. Generalizing the results of
Laver and Woodin, we show that models of ${\rm ZF}+{\rm DC}_\delta$ are
uniformly definable in their set-forcing extensions by posets admitting a gap
at $\delta$, using a ground model parameter. In particular, this means that
models of ${\rm ZF}+{\rm DC}_\delta$ are uniformly definable in their forcing
extensions by posets of size less than $\delta$. We also show that it is
consistent for ground model definability to fail for models of ${\rm ZFC}^-$ of
the form $H_{\kappa^+}$. Using forcing, we produce a ${\rm ZFC}$ universe in
which there is a cardinal $\kappa>\!>\omega$ such that $H_{\kappa^+}$ is not
definable in its Cohen forcing extension. As a corollary, we show that there is
always a countable transitive model of ${\rm ZFC}^-$ violating ground model
definability. These results turn out to have a bearing on ground model
definability for models of ${\rm ZFC}$. It follows from our proof methods that
the hereditary size of the parameter that Woodin used to define a ${\rm ZFC}$
model in its set-forcing extension is best possible.
