We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of $\delta_3^1 = \omega_2$ with $ZFC+AD^{L(R)}$ , and the second proving the uniqueness of the supercompactness measure over ${\cal P}_{\omega_1} (\lambda)$ in L(R) for $\lambda > \delta_1^2$.

We formulate and prove a combinatorial property assuming AD + V = L(R). As a consequence, we show that every regular κ which is either a Suslin cardinal or the successor of a Suslin cardinal is δ 2 1 -supercompact. In particular, all the projective ordinals δ 1 n are δ 2 1 -supercompact.

We formulate and prove a combinatorial property assuming $AD + V = L$. As a consequence, we show that every regular $\kappa$ which is either a Suslin cardinal or the successor of a Suslin cardinal is $\delta^2_1$-supercompact. In particular, all the projective ordinals $\delta^1_n$ are $\delta^2_1$-supercompact.

We show that all the projective ordinals $\delta^1_n$ are supercompact through their supremum $\aleph_{\varepsilon 0}$, and a ways beyond.