Abstract
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 \(\rho _{X}-\rho _{Y}\le 9\).
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Notes
A smooth \(\mathbb {P}^{1}\)-fibration is a smooth morphism with fibers isomorphic to \(\mathbb {P}^{1}\).
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Acknowledgements
I wish to express my thanks to my Ph.D. advisor Cinzia Casagrande for having introduced me to the subject. I am grateful for her constant guidance, for her several and essential suggestions, and for everything I learnt thanks to her. The final part of this work was written during a visit to the Institute of Mathematics of the Polish Academy of Sciences, and it was partially supported by the Grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund. I would to thank IMPAN for the kind hospitality. Finally, I am grateful to Professor St\(\acute{\text {e}}\)phane Druel for some helpful hints.
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Romano, E.A. Non-elementary Fano conic bundles. Collect. Math. 70, 33–50 (2019). https://doi.org/10.1007/s13348-018-0218-x
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DOI: https://doi.org/10.1007/s13348-018-0218-x