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

14H10, 14M20, 14M06, 14H51, 14Q05

unirationality, Hurwitz space