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

TL;DR

This paper investigates the conditions under which a morphism from the tangent bundle of the projective line to a balanced vector bundle arises from a rational curve in projective space, using computational and theoretical methods.

Contribution

It proposes a conjecture linking morphisms to rational curves and provides a computer-assisted proof for specific cases, advancing understanding of rational curves in projective spaces.

Findings

Conjecture relating morphisms to rational curves

Computer-assisted proof for specific (n,d) cases

Insights into the structure of the Syzygy matrix and differential

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 $d$.