Abstract
Let φ:X→W be an elementary divisorial Fano-Mori contraction from a smooth projective variety, defined by a linear system |m(KX+τ L)|, with L a φ-ample line bundle in Pic(X), τ a positive integer and m » 0.
General fibers of such contractions are known to be irreducible if τ≥ dim X−3 (and so if dim X≤4). We prove that, if τ≥ dim X−4, except possibly for one case, a general non trivial fiber is irreducible.
The special case, which can occur when dim X=5, is effctive, as we show by an example in the last section of the paper.
