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

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RESUMO

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 B2 = 1) and the theory of extremal rays (in the case B2 ≥ 2). An n-dimensional smooth projective variety X over k is a Fano manifold if its first Chern class c1(X) ε H2(X, Z) is positive in the sense of Kodaira [Kodaira, K. (1954) Ann. Math. 60, 28-48] (or ample). If n = 3 and c1(X) generates H2(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 codimGX 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 X1210 [unk] Pk15, or (iii) X is isomorphic to a double cover of Pk3, a 3-dimensional quadric Qk3, or a quintic del Pezzo 3-fold V5 [unk] Pk6. If n = 4 and c1(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 P1 and a Fano 3-fold, (c) the blow-up of Q4 [unk] P5 along a line or along a conic, or (d) a P1-bundle compactifying a line bundle on P3 or on Q3 [unk] P4.

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