A note on gonality of curves on general hypersurfaces

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces (Formula presented.). 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 (Formula presented.) is a very general hypersurface of degree (Formula presented.), the least gonality of a curve (Formula presented.) passing through a general point of X is (Formula presented.), apart from some exceptions we list