Gonality of curves on general hypersurfaces

TL;DR

This paper determines the minimal gonality of curves passing through a general point on very general hypersurfaces of high degree, refining previous bounds and identifying specific gonality values with some exceptions.

Contribution

It establishes an exact formula for the least gonality of such curves on very general hypersurfaces of degree at least 2n+2, improving prior bounds.

Findings

Exact gonality formula for very general hypersurfaces

Gonality bounds improved for high-degree hypersurfaces

Identification of possible exceptions in gonality values

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 $d\geq n+2$, and if $C\subset X$ is an irreducible curve passing through a general point of $X$, then its gonality verifies $\mathrm{gon}(C)\geq d-n$, 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 $d\geq 2n+2$, the least gonality of an irreducible curve $C\subset X$ passing through a general point of $X$ is $\mathrm{gon}(C)=d-\left\lfloor\frac{\sqrt{16n+1}-1}{2}\right\rfloor$, apart from a series of possible exceptions, where $\mathrm{gon}(C)$ may drop by one.