A note on gonality of curves on general hypersurfaces (original) (raw)

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces X ⊂ P n+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 2n + 2, the least gonality of a curve C ⊂ X passing through a general point of X is gon(C) = d − √ 16n+1−1 2 , apart from some exceptions we list. The research leading to these results has received funding from the European Research Council under the