Higher order minimal families of rational curves and Fano manifolds with nef Chern characters

Abstract

In this paper, we investigate higher order minimal families 𝐻_𝑖 of rational curves associated to Fano manifolds 𝑋. We prove that 𝐻_𝑖 is also a Fano manifold if the Chern characters of 𝑋 satisfy some positivity conditions. We also provide a sufficient condition for Fano manifolds to be covered by higher rational manifolds.