On the Picard number of divisors in Fano manifolds

Details On the Picard number of divisors in Fano manifolds On the Picard number of divisors in Fano manifolds

2009-05-20

arxiv.org/abs/0905.3239v4

Mathematics

Algebraic Geometry

Final version, to appear in the Annales Scientifiques de l'Ecole Normale Superieure

Scientific paper

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 with pseudo-index >1, and to surjective morphisms whose source is Fano, having some high-dimensional fibers or low-dimensional target.

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