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Let $\Gamma$ be a smooth $k$-dimensional hypersurface in $\mathbb P^{k+1}$and $X \supset \Gamma$ a smooth $n$-dimensional Fano hypersurface in $\mathbbP^{n+1}$ 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 dreamspace. In particular, we describe its Mori chamber decomposition and theassociated birational models of $Y$. We classify for which $X$ and $\Gamma$ thevariety $Y$ is a Fano manifold and we initiate the study of K-stability of $Y$.

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