Gonality of curves on general hypersufaces

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