On Fano manifolds with an unsplit dominating family of rational curves

TL;DR

This paper investigates Fano manifolds with specific rational curve families, proving the Generalized Mukai Conjecture holds under certain conditions related to the manifold's dimension and pseudoindex.

Contribution

It establishes the validity of the Generalized Mukai Conjecture for Fano manifolds with pseudoindex equal to one-third of the dimension or in six dimensions, extending known results.

Findings

The conjecture holds if pseudoindex i_X = (dimension)/3.

The conjecture is true for all Fano manifolds with i_X > (dimension)/3.

The conjecture is verified in dimension 6.

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 $i_X > (\dim X)/3$.