A note on gonality of curves on general hypersurfaces

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces X⊂ Pn+1. After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if X⊂P^{n+1} is a very general hypersurface of degree d⩾ 2 n+ 2, the least gonality of a curve C⊂X passing through a general point of X is gon(C)=d-[16n+1-12], apart from some exceptions we list.