Non-uniruledness results for spaces of rational curves in hypersurfaces

TL;DR

This paper investigates the geometric properties of spaces of rational curves in hypersurfaces, proving they are not uniruled under certain degree conditions, which advances understanding of their structure in algebraic geometry.

Contribution

It establishes new non-uniruledness results for spaces of rational curves in hypersurfaces, extending previous knowledge about their geometric complexity.

Findings

Spaces of rational curves are not uniruled if (n+1)/2 ≤ d ≤ n-3.

For any degree e, the space of rational curves of degree e is not uniruled if d ≥ e√n.

Results apply to smooth hypersurfaces in projective space.

Abstract

We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree $d$ in $P^n$ are not uniruled if $(n+1)/2 \leq d \leq n-3$. We also show that for any positive integer $e$, the space of smooth rational curves of degree $e$ in a general hypersurface of degree $d$ in $P^n$ is not uniruled when $d \geq e \sqrt{n}$.