Geometry of chains of minimal rational curves

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.