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The blow-up of the cubic fourfold along eight planes is Fano
We prove that the blow-up Z_m of a smooth cubic fourfold along m <= 8 planes, pairwise meeting at distinct points and satisfying explicit generality conditions, is a smooth Fano fourfold of Picard rank rho(Z_m) = m+1 <= 9; in particular, we obtain three new families of smooth Fano fourfolds with rho
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