The unirationality of Hurwitz spaces of 6-gonal curves of small genus

Abstract

In this short note we prove the unirationality of Hurwitz spaces of 6-gonal curves of genus $g$ with $5\le g\le 28$ or $g=30,31,35,36,40,45$. Key ingredient is a liaison construction in ${\mathbb{P}}^1\times {\mathbb{P}}^2$. By semicontinuity, the proof of the dominance of this construction is reduced to a computation of a single curve over a finite field.