On the Picard number of divisors in Fano manifolds

TL;DR

This paper investigates the Picard number of divisors in Fano manifolds, establishing bounds on the codimension of certain cycle images and characterizing the structure of manifolds when these bounds are exceeded.

Contribution

It provides a new upper bound on the codimension of the image of 1-cycles of a divisor in a Fano manifold and classifies the structure of manifolds when this bound is surpassed.

Findings

The codimension c of H in N_1(X) is at most 8.

If c>2, then X is either a product involving a Del Pezzo surface or admits a Del Pezzo fibration with specific Picard number properties.

Applications include classifications and properties of Fano 4-folds and related morphisms.

Abstract

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.