The blow-up of the cubic fourfold along eight planes is Fano

Abstract

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 > 6 that are not a product of surfaces. Complementary to this result, we show that a smooth cubic fourfold as above exists. Moreover, we study the birational geometry of Z_m and show that it corresponds exactly to that of a smooth del Pezzo surfaces S_m, obtained as the blow-up of P^2 along m <= 8 points in general position. Namely, Z_m and S_m have isomorphic Mori cones, pseudo-effective cones with same Mori chamber decomposition, and there is a one-to-one correspondence between the contractions of Z_m and the contractions of S_m. Lastly, we study the anticanonical linear system of Z_m and prove that it is basepoint free if and only if m <= 7. For m = 8 and a special configuration of planes, we show that the anticanonical base locus Bs(|-K_{Z_8}|) has dimension 2 and contains a smooth del Pezzo surface of degree 1.