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We give an explicit parametrization of the Hilbert schemes of rational curvesC in P^n having a given splitting type of the restricted tangent bundle fromP^n to C. The adopted technique uses the description of such curves asprojections of a rational normal curve from a suitable linear vertex and aclassification of those vertices that correspond to the required splitting typeof the restricted tangent bundle. This classification involves the study of asuitable PGL(2) action on the relevant Grassmannian variety.

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