LAUSR

Abstract Let n ≥ 4 , 2 ≤ r ≤ n − 2 and e ≥ 1 . We show that the intersection of the locus of degree e morphisms from 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 along at least one component of the intersection. As a consequence, we get that the locus of degree e morphisms from P 1 to G ( r , n ) with the restricted tangent bundle having a given splitting type need not always be irreducible.