Blowups of smooth hypersurfaces, their birational geometry and divisorial stability

Abstract

Let $X$ be a smooth $n$-dimensional Fano hypersurface in $\mathbb P^{n+1}$ where $n \geq 3$. Let $\Gamma$ be a smooth positive-dimensional complete intersection of $X$, a hypersurface and one of more hyperplanes in $\mathbb P^{n+1}$. Let $Y \to X$ be the blowup of $X$ along $\Gamma$. Let $\varphi \colon Y \rightarrow X$ be the blowup of $X$ along $\Gamma$. We describe the Mori chamber decomposition of $Y$ and its associated birational models. In particular, we show that $Y$ is a Mori dream space. We classify for which $X$ and $\Gamma$ the variety $Y$ is a Fano manifold and, if $X$ is a hyperplane, we classify the elementary Sarkisov links initiated by $\varphi$. Finally, we use this Mori chamber decomposition above to prove that certain Fano manifolds as above do not admit a K\"ahler-Einstein metric.