Coskun–Harris–Starr conjecture on rational curves on general Fano hypersurfaces

Let $X\subset \mathbb{P}^n$ be a general hypersurface of degree $d\leq n$ and dimension at least $3$. Let $R_e(X)$ be the open locus in the Hilbert scheme of $X$ parameterizing smooth rational curves of degree $e$, and let $\overline{\mathcal{M}}_{0,1}(X,e)$ denote the moduli space of stable maps with one marked point. Coskun–Harris–Starr conjecture. The locus $R_e(X)$ is irreducible of dimension

$$e(n+1-d)+n-4.$$

Furthermore, if $d\leq n-1$, the evaluation map

$$\overline{\mathcal{M}}_{0,1}(X,e)\to X$$

is flat. This conjecture predicts the expected geometry of rational curves on general Fano hypersurfaces; the paper proves additional cases, while the full statement is not established in the supplied text.