Let X be a smooth n-dimensional Fano hypersurface in Pⁿ⁺¹ where n is greather than or equal to 3. Let Γ be a smooth positive-dimensional complete intersection of X, a hypersurface and one of more hyperplanes in Pⁿ⁺¹. Let Y -> X be the blowup of X along Γ. Let ϕ: Y -> 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 Γ the variety Y is a Fano manifold and, if X is a hyperplane, we classify the elementary Sarkisov links initiated by ϕ. Finally, we use this Mori chamber decomposition above to prove that certain Fano manifolds as above do not admit a Kahler-Einstein metric.

The first author was previously supported by the Korea Institute for Advanced Study (KIAS), grant No. MG087901. The second author was supported by Engineering and Physical Sciences Research Council (EPSRC) EP/V055399/1 and ERC StG Saphidir No. 101076412. The third author was supported by the Simons Investigator Award HMS, National Science Fund of Bulgaria, National Scientific Program "Excellent Research and People for the Development of European Science" (VIHREN), Project No. KP-06-DV-7 and this is a contribution to Project-ID 286237555 -- TRR 195 -- by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).