Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\subseteq \mathbb{R}$ uncountable and OD. Clearly if $X$ has an OD perfect subset, then $\theta_0(X)=\theta_0$ (the first term in the Solovay sequence); howevr, this need not occur, as Andreas Blass' answer to Ordinal-definable witnesses to the perfect set property? shows (consider the set of non-OD reals).

My question is: is it consistent with ZF+AD that there is an uncountable OD set of reals $X$ with $\theta_0(X)<\theta_0$? If so, what can we say about the possible values of the $\theta_0$-map?

My motivation is, ultimately, the question Cardinal characteristics without choice I'm especially interested in cardinal characteristics which have natural "$\kappa$-versions" for arbitrary $\kappa$ (e.g. the idea of the dominating number makes sense on $\kappa^\kappa$, not just $\omega^\omega$), and the possibility of comparing such cardinal characteristics by looking at the Solovay-type sequences their "large" sets induce; but that's jumping the gun by a substantial amount. I mention this here only in case this motivation is helpful re: finding references, etc.

Working in ZF+AD, let $$\theta_0(X)=\min\{\alpha\in ON: \not\exists f: X\rightarrow \alpha\mbox{ surjective and OD}\}$$ be the least ordinal onto which $X$ does not surject in an OD way, for $X\subseteq \mathbb{R}$ uncountable and OD. Clearly if $X$ has an OD perfect subset, then $\theta_0(X)=\theta_0$ (the first term in the Solovay sequence); howevr, this need not occur, as Andreas Blass' answer to Ordinal-definable witnesses to the perfect set property? shows (consider the set of non-OD reals).

My question is: is it consistent with ZF+AD that there is an uncountable OD set of reals $X$ with $\theta_0(X)<\theta_0$? If so, what can we say about the possible values of the $\theta_0$-map?

My motivation is, ultimately, the question Cardinal characteristics without choice I'm especially interested in cardinal characteristics which have natural "$\kappa$-versions" for arbitrary $\kappa$ (e.g. the idea of the dominating number makes sense on $\kappa^\kappa$, not just $\omega^\omega$), and the possibility of comparing such cardinal characteristics by looking at the Solovay-type sequences their "large" sets induce; but that's jumping the gun by a substantial amount. I mention this here only in case this motivation is helpful re: finding references, etc.