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 f:X\rightarrow Y. Denoting by \rho _{X} the Picard number of X, we investigate such contractions when \rho _{X}-\rho _{Y}>1, called non-elementary. We prove that \rho _{X}-\rho _{Y}\le 8, and we deduce new geometric information about our varieties X and Y, depending on \rho _{X}-\rho _{Y}. 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\rightarrow Y is a fiber type K_{X}-negative contraction with one-dimensional fibers, we show that \rho _{X}-\rho _{Y}\le 9.