Biregular classification of Fano 3-folds and Fano manifolds of coindex 3

The Fano 3-folds and their higher dimensional analogues are classified over an arbitrary field k unk C by applying the theory of vector bundles (in the case B(2) = 1) and the theory of extremal rays (in the case B(2) >/= 2). An n-dimensional smooth projective variety X over k is a Fano manifold if its first Chern class c(1)(X) epsilon H(2)(X, Z) is positive in the sense of Kodaira Kodaira, K. (1954) Ann. Math. 60, 28-48 (or ample). If n = 3 and c(1)(X) generates H(2)(X, Z), then either (i) X is a complete intersection in a Grassmann variety G with respect to a homogeneous vector bundle E on G: the rank of E is equal to codim(G)X and X is isomorphic to the zero locus of a global section of E, (ii) X is a linear section of a 10-dimensional spinor variety X(12) (10) unk P(k) (15), or (iii) X is isomorphic to a double cover of P(k) (3), a 3-dimensional quadric Q(k) (3), or a quintic del Pezzo 3-fold V(5) unk P(k) (6). If n = 4 and c(1)(X) is divisible by 2, then X unk C is isomorphic to (a) a complete intersection in a homogeneous space or its double cover, (b) a product of P(1) and a Fano 3-fold, (c) the blow-up of Q(4) unk P(5) along a line or along a conic, or (d) a P(1)-bundle compactifying a line bundle on P(3) or on Q(3) unk P(4).