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

TL;DR

This paper provides an explicit parametrization of Hilbert schemes of rational curves in projective space with specified tangent bundle splitting, using PGL(2) actions on Grassmannians and projections of rational normal curves.

Contribution

It introduces a new method to classify rational curves with given tangent bundle splitting via PGL(2) actions and projections, offering explicit parametrizations.

Findings

Explicit parametrization of Hilbert schemes for rational curves

Classification of vertices related to tangent bundle splitting

Use of PGL(2) actions on Grassmannians for curve analysis

Abstract

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