Rational curves of degree 16 on a general heptic fourfold (Q392388)

The author studies rational curves on a (very) general degree \(7\) hypersurface in \(\mathbb{P}^5\). He proves that the only rational curves of degree at most \(16\) that lie on such a hypersurface are lines. This result builds on previous work of the author, extends work of \textit{T. Johnsen} and \textit{G. M. Hana} [Bull. Belg. Math. Soc. - Simon Stevin 16, No. 5, 861--885 (2009; Zbl 1183.14059)], and proves a special case of a conjecture made by H. Clemens.NEWLINENEWLINEMore specifically, Hana-Johnsen had established this result for rational curves of degree at most \(15\). The main effort of the present paper is thus is to extend this result to rational curves of degree \(16\).NEWLINENEWLINESome interesting parts to the proof of this results include the authors use of generic initial ideals, Castelnuovo's bound, the uniform position principle, and the computer algebra system Macaulay2.