$\text{NS}_{\omega_1}$ saturated, $\Delta_1 ( \{ \omega_1 \} )$-definable and a $\Delta^1_4$-definable well-order of the reals

Abstract

Assuming $M_1$, the canonical inner model with one Woodin cardinal exists, we construct a model in which the nonstationary ideal on $\omega_1$ is $\aleph_2$-saturated, $\Delta_1$-definable with $\omega_1$ as the only parameter and there is a $\Sigma^1_{4}$-definable well-order of the reals. This implies that contrary to the assumption that $NS_{\omega_1}$ is $\aleph_1$-dense, the assumption of $NS_{\omega_1}$ being saturated and $\Delta_1$-definable does not imply any nice structural properties for the projective subsets of the reals.