  Let $n\geq 4$, $2 \leq r \leq n-2$ and $e \geq 1$. We show that the
intersection of the locus of degree $e$ morphisms from $\mathbb{P}^1$ to
$G(r,n)$ with the restricted universal sub-bundles having a given splitting
type and the locus of degree $e$ morphisms with the restricted universal
quotient-bundle having a given splitting type is non-empty and generically
transverse. As a consequence, we get that the locus of degree $e$ morphisms
from $\mathbb{P}^1$ to $G(r,n)$ with the restricted tangent bundle having a
given splitting type need not always be irreducible.

Algebraic Geometry