A note on rational curves on general Fano hypersurfaces

TL;DR

This paper proves that on general Fano hypersurfaces, the space of rational curves of certain degrees is well-behaved, confirming parts of a conjecture and showing Gromov-Witten invariants are enumerative.

Contribution

It establishes the structure and dimension of the Kontsevich space of rational curves on general Fano hypersurfaces, advancing understanding of their enumerative geometry.

Findings

The Kontsevich space is equidimensional of expected dimension.

It has two components: smooth rational curves and multiple covers of a line.

Gromov-Witten invariants are shown to be enumerative in these cases.

Abstract

We show the Kontsevich space of rational curves of degree at most roughly $\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^n$ of degree $n-1$ is equidimensional of expected dimension and has two components: one consisting generically of smooth, embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these cases are enumerative.