Resumen de Non-elementary Fano conic bundles

Emmanuel A. Romano

• Let X be a complex, projective, smooth and Fano variety. We study Fano conic bundles ?:?→? . Denoting by ?? the Picard number of X, we investigate such contractions when ??−??>1 , called non-elementary. We prove that ??−??≤8 , and we deduce new geometric information about our varieties X and Y, depending on ??−?? . 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 ?:?→? is a fiber type ?? -negative contraction with one-dimensional fibers, we show that ??−??≤9 .