Inner models and ultrafilters in L($\mathbb{R})

Abstract

We present a characterization of supercompactness measures for $\omega_1$ in L($\mathbb{R}$), 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 $\mathsf{ZFC+AD}^{L\mathbb{R}}$, and the second proving the uniqueness of the supercompactness measure over $\mathcal{P}_{\omega_l }(\lambda)$ in L ($\mathbb{R}$) for $\lambda > \delta_1^2$.