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

Let V congruent to 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) on any Grassmannian 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 >= 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 subset of P-s with s <= 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. (C) 2014 Elsevier B.V. All rights reserved.