Locally unsplit families of rational curves of large anticanonical   degree on Fano manifolds
  In this paper we address Fano manifolds X with a locally unsplit dominating family of rational curves of anticanonical degree equal to the dimension of X. We first observe that their Picard number is at most 3, and then we provide a classification of all cases with maximal Picard number. We also give examples of locally unsplit dominating families of rational curves whose varieties of minimal tangents at a general point is singular. 
Cinzia Casagrande and St\'ephane Druel
math.AG