On the Picard number of divisors in Fano manifolds.
(Sur le nombre de Picard des diviseurs dans les variétés de Fano.)

*(English. French summary)*Zbl 1267.14050Let \(X\) be a Fano manifold and \(D\) a prime divisor in \(X\). Denote by \(\rho_X\) and \(\rho_D\) the Picard numbers of \(X\) and \(D\), respectively, and by \(c(D)\) the codimension of the image of \(N_1(D)\) in \(N_1(X)\) under the natural push-forward of \(1\)-cycles. If \(X\) is a surface then \(c(D)=\rho_X-1 \leq 8\); the main result of the paper under review states that this hold in any dimension, i.e. we have
\[
\rho_X-\rho_D \leq c(D) \leq 8.
\]
Moreover, the existence of a divisor with large \(c(D)\) provides strong conditions on the variety \(X\), namely, if \(c(D) \geq 3\) either \(X\) is a product with a surface as a factor, or \(c(D)=3\) and \(X\) admits a flat surjective morphism with 2-dimensional connected fibers onto a Fano manifold \(T\) with \(\rho_T=\rho_X-4\). Both cases are effective.

This result can be applied in order to get bounds on the Picard number of \(X\) to Fano manifolds of dimension \(\leq 5\) and to Fano manifolds having a morphism to a curve.

Moreover a stronger version of the above theorem can be obtained for Fano manifolds of pseudoindex \(i_X \geq 2\).

The main idea occurring in the (long and well-organized) proof is the use of a Mori program for \(-D\), where \(D\) is a prime divisor (the existence of such a program has been proved in [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]) which is special, meaning that all the involved extremal rays have positive intersection with the anticanonical divisor.

This result can be applied in order to get bounds on the Picard number of \(X\) to Fano manifolds of dimension \(\leq 5\) and to Fano manifolds having a morphism to a curve.

Moreover a stronger version of the above theorem can be obtained for Fano manifolds of pseudoindex \(i_X \geq 2\).

The main idea occurring in the (long and well-organized) proof is the use of a Mori program for \(-D\), where \(D\) is a prime divisor (the existence of such a program has been proved in [J. Am. Math. Soc. 23, No. 2, 405–468 (2010; Zbl 1210.14019)]) which is special, meaning that all the involved extremal rays have positive intersection with the anticanonical divisor.

Reviewer: Gianluca Occhetta (Trento)