PGL(2)PGL(2) actions on Grassmannians and projective construction of rational curves with given restricted tangent bundle

Let V≅Cd+1V≅Cd+1 be a complex vector space. If one identifies it with a space of binary forms of degree d  , then one gets an action of PGL(2)PGL(2) on any Grassmannian Gr(e+1,V)Gr(e+1,V). We will produce some refined numerical invariants for such an action that stratify the Grassmannian into irreducible and rational invariant strata. Assuming d≥3d≥3, the numerical invariants so obtained are shown to correspond in a simple way with the set of possible splitting types of the restricted tangent bundles of degree d   rational curves C⊂PsC⊂Ps with s≤d−1s≤d−1. By means of the same techniques we produce explicit parameterizations for the varieties of rational curves with a given splitting type of the restricted tangent bundle.