Fano manifolds and quadric bundles (Q1323442)

Here the author proves (using Mori theory, the deformations of rational curves, the so-called Mori's breaking technique and results on uniform bundles) the following theorem. Let \(X\) be a Fano manifold of index \(r\) and dimension \(2r\) (characteristic 0). Assume \(r \geq 3\) and \(b_ 2(X) \geq 2\). Then \(X\) is either a projective bundle over a Fano manifold of dimension \(r + 1\) or it is a quadric bundle over a smooth variety or \(X\) has two elementary contractions such that either each contraction makes \(X\) a non- equidimensional scroll or one of the contractions makes \(X\) a non- equidimensional scroll and the other is birational and of divisorial type. In an added in proof the author describes a later result (a complete classification of Fano manifolds of index \(r\), dimension \(2r\) and \(b_ 2 \geq 2)\) contained in a joint paper of the reviewer and the author [``On Banica sheaves and Fano manifolds'', Compos. Math. (to appear)].