Fano-Mori elementary contractions with reducible general fiber

Abstract

Let φ: X → W be an elementary divisorial Fano-Mori contraction from a smooth projective variety, defined by a linear system |m(K_X + τL)|, 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 effective, as we show by an example in the last section of the paper.