On Fano manifolds with an unsplit dominating family of rational curves

Abstract

We study Fano manifolds X admitting an unsplit dominating family of rational curves and we prove that the Generalized Mukai Conjecture holds if X has pseudoindex i_X = (dim X)/3 or dimension dim X = 6. We also show that this conjecture is true for all Fano manifolds with i_X > (dim X)/3.