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⊂P^{n+1} is a hypersurface of degree d⩾n+2, and if C⊂X is an irreducible curve passing through a general point of X, then its gonality verifies gon(C)⩾d−n, and equality is attained on some special hypersurfaces. We prove that if X⊂P^{n+1} is a very general hypersurface of degree d⩾2n+2, the least gonality of an irreducible curve C⊂X passing through a general point of X is gon(C)=d−[(sqrt(16n+1)-1)/2], apart from a series of possible exceptions, where gon(C) may drop by one.