Gonality of curves on general hypersurfaces

Abstract

This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if $X\subset {\mathbb{P}}^{n+1}$ is a hypersurface of degree $d\ge n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{gon}(C)\ge d-n$, and equality is attained on some special hypersurfaces. We prove that if $X\subset {\mathbb{P}}^{n+1}$ is a very general hypersurface of degree $d\ge 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general point of $X$ is $\mathrm{gon}(C)=d-\lfloor \frac{\sqrt{16n+1}-1}{2}\rfloor$, apart from a series of possible exceptions, where $\mathrm{gon}(C)$ may drop by one.

24 pages. We corrected various inaccuracies pointed out by the referee

References in corpus (2)

• Measures of irrationality for hypersurfaces of large degree
• On irrationality of surfaces in ${\mathbb{P}}^3$

Cited by in corpus (1)

• Covering gonalities of complete intersections in positive characteristic