The Syzygy Matrix and the Differential for Rational Curves in Projective Space

Abstract

In this paper, we study whether a given morphism $f$ from the tangent bundle of ${\mathbb{P}}^1$ to a balanced vector bundle of degree $(n+1)d$ is induced by the restriction of the tangent bundle $T_{{\mathbb{P}}^n}$ to a rational curve of degree $d$ in ${\mathbb{P}}^n$. We propose a conjecture on this problem based on Mathematica computations of some examples and provide computer-assisted proof of the conjecture for certain values of $n$ and $d$.