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 in IP^{n+1} is a hypersurface of degree d > n+1, and if C is an irreducible curve in X passing through a general point of X, then its gonality verifies gon(C) >= d-n, and equality is attained on some special hypersurfaces. Under the assumption of X a very general hypersurface of degree d >= 2n+2, we compute the least gonality c of an irreducible curve C in X passing through a general point of X apart from a series of possible exceptions, where the gonality may drop by one.