The Unirationality of Hurwitz Spaces of 6-Gonal Curves of Small Genus

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.

2010 Mathematics Subject Classification: 14H10, 14M20, 14M06, 14H51, 14Q05

Keywords and Phrases: Unirationality, Hurwitz space.

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