Non-elementary Fano conic bundles

Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundlesf: X→ Y. Denoting by ρX the Picard number of X, we investigate such contractions when ρX- ρY> 1 , called non-elementary. We prove that ρX- ρY≤ 8 , and we deduce new geometric information about our varieties X and Y, depending on ρX- ρY. Using our results, we show that some known examples of Fano conic bundles are elementary. Moreover, when we allow that X is locally factorial with canonical singularities and with at most finitely many non-terminal points, and f: X→ Y is a fiber type KX-negative contraction with one-dimensional fibers, we show that ρX- ρY≤ 9.