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

Let V be a (d+1)-dimensional complex vector space. If one identifies it with a space of binary forms of degree d, then one gets an action of PGL(2) on any Grassmannian G(e+1,V). We will produce some refined numerical invariants for such an action that stratify G(e+1,V) into irreducible and rational invariant strata. Assuming d≥3, the invariants so obtained are shown to correspond with the set of possible splitting types of the restricted tangent bundles of degree d rational curves in P^s with s ≤d-1.