Blowups of smooth Fano hypersurfaces, their birational geometry and divisorial stability

Let $\Gamma$ be a smooth $k$-dimensional hypersurface in $\mathbb P^{k+1}$ and $X \supset \Gamma$ a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n\geq 3$ and $k\geq 1$. Let $Y \rightarrow X$ be the blowup of $X$ along $\Gamma$. We give a constructive proof that $Y$ is a Mori dream space. In particular, we describe its Mori chamber decomposition and the associated birational models of $Y$. We classify for which $X$ and $\Gamma$ the variety $Y$ is a Fano manifold and we initiate the study of K-stability of $Y$.