On the Picard number of Fano 6-folds with a non-small contraction

Abstract

A generalization of S. Mukai's conjecture says that if $X$ is a Fano $n$-fold with Picard number ${\rho }_X$ and pseudo-index $i_X$, then ${\rho }_X(i_X-1)\le n$, with equality if and only if $X\cong ({\mathbb{P}}^{i_X-1}{)}^{{\rho }_X}$. In this paper, we prove that this conjecture holds if $n=6$ and either $X$ admits a contraction of fiber type or $X$ admits no small contractions.

16 pages