Restricted tangent bundle of rational curves on projective hypersurfaces

TL;DR

This paper classifies when general projective hypersurfaces contain rational curves with balanced restricted tangent bundles and provides methods to explicitly construct such examples, advancing understanding of curve geometry on hypersurfaces.

Contribution

It determines all triples (e, d, n) for which general hypersurfaces contain rational curves with balanced tangent bundles and offers explicit construction techniques.

Findings

Classifies all (e, d, n) triples for balanced tangent bundles on rational curves.

Provides explicit examples of hypersurfaces with balanced restricted tangent bundles.

Enhances understanding of the geometry of rational curves on hypersurfaces.

Abstract

We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset \mathbb{P}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X|_C$. In addition, we show how to compute explicit examples of hypersurfaces with balanced $T_X|_C$ when $C$ is a rational normal curve.