If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the degrees of constructibility $\mathcal{C}$. This poset is extremely dependent on the ambient set theory. If $V=L$, the poset is trivial, whereas the existence of a Cohen real automatically leads to a very complicated structure (see, e.g., the paper "The degrees of constructibility of Cohen reals" by Abraham and Shore).
My question is basically, what is known about the possible posets which $\mathcal{C}$ could be? I am especially interested in properties of $\mathcal{C}$ which follow from large cardinals. For example, if $0^\sharp$ exists, then we can deduce that $\mathcal{C}$ has size $2^{\aleph_0}$, which is the maximum possible; presumably this actually gives us a lot more.
As a sub-question, what is a good source for learning about techniques for building a model of $ZFC$ in which $\mathcal{C}$ has some prescribed properties?
ADDED: I'm most interested in what can be said when the size of $\mathcal{C}$ is assumed to be $2^{\aleph_0}$. As a particular example, the paper "Hinges and automorphisms of the degrees of non-constructibility" by P. Farrington (http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.96.3272) has, as Lemma 2.5, the following (CAVEAT: Farrington's paper actually gives the wrong definition of "hinge," which was corrected in Lubarsky's review in the JSL in 1989):
Suppose that $\omega_2^L<\omega_1$. Then there is no nontrivial automorphism of $\mathcal{C}$.
(Now $\omega_2^L<\omega_1$ (and much more!) is a consequence of the existence of $0^\sharp$, so this is a very weak hypothesis.) Results like this are exactly what I'm looking for: properties $\mathcal{C}$ which follow from large cardinals, and in particular can hold under the assumption $\vert\mathcal{C}\vert=2^{\aleph_0}$.
