Rational curves and bounds on the Picard number of Fano manifolds

Abstract

We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X\ge (\dim X+3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.

arXiv commentModified the proof of the main theorem and eliminated a statement about the cone of curves. To appear in Geometriae Dedicata