Assume that the Axiom of Choice holds in $V$, the von Neumann universe. As Andreas Blass explained, $\mathsf{DC}$ holds in $L(\mathbb R)$.
Is this the case for $\mathsf{AC}_{\omega_1}$ too? That is, do we have $L(\mathbb R)\models \mathsf{AC}_{\omega_1}$?
(Here $\mathsf{AC}_{\omega_1}$ stands for the Axiom of Choice for families of cardinality $\omega_1$.)
It seems to me that the answer is yes, as $\mathsf{AC}_{\omega_1}$ involves only bounded quantifiers. Is this right?
No.
For example in Solovay's model, we start with $L$ and an inaccessible cardinal $\kappa$, and then consider $L(\Bbb R)$ in $L[G]$, where $G$ is $L$-generic for collapsing $\kappa$ to be $\omega_1$.
In this model there is no set of size $\aleph_1$ of reals, but there is always a surjection from $\Bbb R$ onto $\omega_1$. So $\sf AC_{\omega_1}$ must fail.
Other examples come from the natural models of $\sf AD$, which of course also disproves $\sf AC_{\omega_1}$. I have a sneaky suspicion that the large cardinal is necessary here, but I don't see any solution one way or another.
I will note that the "standard attempt" of adding $\omega_1$ Cohen reals and looking at $L(\Bbb R)$ will satisfy $\sf AC_{WO}$, that is choice from families which can be well-ordered, so in particular $\sf AC_{\omega_1}$ (this also gives an alternative proof of $\sf DC$).