Rational curves on blowing-ups of projective spaces
  We prove the irredcibility (and the rational connectedness) of the moduli spaces of (free) morphisms from a projective line to a successive blowing-up of a product of projective spaces if a suitable numerical condition on morphisms is provided (and also if all centers of the blowing-up are rationally connected). 
Bumsig Kim, Yongnam Lee and Kyungho Oh