MR 3007673 Reviewed Geiss F. The Unirationality of Hurwitz spaces of 6-gonal curves of small genus. Documenta Mathematica (2012) 17, 627--640. (Reviewer Francesca Vetro) 14H10 (14M20)

Let H (d, w) be the Hurwitz space that parametrizes degree d simple coverings of the projective line with w = 2g + 2d - 2 branch points. A classic result affirms the unirationality of these spaces for d \leq 3. Successively, Arbarello and Cornalba in [E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341--362] prove that the spaces H (d, w) are unirational in the following cases: d \leq 5 and g \geq d - 1, d = 6 and 5 \leq g \leq 10 or g = 12 and d = 7 and g = 7. In this paper, the author studies the problem of unirationality over an algebraically closed field of characteristic zero when d = 6. In particular, the author proves that the spaces H (6, 2g + 10) are unirational for 5 \leq g \leq 28 or g = 30, 31, 33, 35, 36, 40, 45. The proof of this result is based on the observation that a general 6-gonal curve in P^1 x P^2 can be linked to the union of a rational curve and a collection of lines.