The Baire and perfect set properties at singulars cardinals

We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa +1})$ has both the $\kappa$-Perfect Set Property and the $\vec{\mathcal{U}}$-Baire Property. This is a higher analogue of Solovay’s result for $L(\mathbb{R})$. We obtain this configuration starting with large-cardinal assumptions in the realm of supercompactness, thus improving former theorems by Cramer, Shi and Woodin.