Rational curves of degree 16 on a general heptic fourfold

TL;DR

This paper proves that a general heptic fourfold in projective 5-space contains no rational curves of degree 16, supporting Clemens' conjecture for this case and extending previous results up to degree 15.

Contribution

The authors establish the non-existence of degree 16 rational curves on a general heptic fourfold, extending prior work that confirmed this for degrees up to 15.

Findings

No rational curves of degree 16 on a general heptic fourfold

Supports Clemens' conjecture in this specific case

Extends previous degree bounds from 15 to 16

Abstract

According to a conjecture of H. Clemens, the dimension of the space of rational curves on a general projective hypersurface should equal the number predicted by a na\"ive dimension count. In the case of a general hypersurface of degree 7 in $\mathbb{P}^5$, the conjecture predicts that the only rational curves should be lines. This has been verified by Hana and Johnsen for rational curves of degree at most 15. Here we extend their results to show that no rational curves of degree 16 lie on a general heptic fourfold.