Gonality of curves on general hypersurfaces

This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset {\mathbb{P}}^{n+1}$ is a hypersurface of degree $d\ge n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{g}\mathrm{o}\mathrm{n}(C)\ge d-n$, and equality is attained on some special hypersurfaces. We prove that if $X\subset {\mathbb{P}}^{n+1}$ is a very general hypersurface of degree $d\ge 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general point of $X$ is $\mathrm{g}\mathrm{o}\mathrm{n}(C)=d-\lfloor \frac{\sqrt{16n+1}-1}{2}\rfloor$, apart from a series of possible exceptions, where $\mathrm{g}\mathrm{o}\mathrm{n}(C)$ may drop by one.