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.

Non-elementary Fano conic bundles / Romano, E. A.. - In: COLLECTANEA MATHEMATICA. - ISSN 0010-0757. - 70:1(2019), pp. 33-50. [10.1007/s13348-018-0218-x]

Non-elementary Fano conic bundles

Romano E. A.
2019-01-01

Abstract

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.
2019
1
Romano, E. A.
Non-elementary Fano conic bundles / Romano, E. A.. - In: COLLECTANEA MATHEMATICA. - ISSN 0010-0757. - 70:1(2019), pp. 33-50. [10.1007/s13348-018-0218-x]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/277719
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