On the Picard number of divisors in Fano manifolds

@article{Casagrande2009OnTP,
  title={On the Picard number of divisors in Fano manifolds},
  author={C. Casagrande},
  journal={arXiv: Algebraic Geometry},
  year={2009}
}
  • C. Casagrande
  • Published 2009
  • Mathematics
  • arXiv: Algebraic Geometry
  • Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image H of N_1(D) in N_1(X) under the natural push-forward of 1-cycles. We show that the codimension c of H in N_1(X) is at most 8. Moreover if c>2, then either X=SxY where S is a Del Pezzo surface, or c=3 and X has a flat fibration in Del Pezzo surfaces onto a Fano manifold Y, such that the difference of the Picard numbers of X and Y is 4. We give applications to Fano 4-folds, to Fano varieties… CONTINUE READING
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