On Σ-Hilbertian fields

A held K is 0-Hilbertian if K not equal U-i=1(n) phi(i)(K) for any collection of rational functions cpi of degree at least 2, i = I,..., m. Corvaja and Zannier [CoZ] give an elementary construction for a 0-Hilbertian field that isn't Hilbertian. There is an obvious generalization of the notion of 0-Hilbertian to g-Hilbertian. Guralnick-Thompson and Liebeck-Saxl have given a partial classification of monodromy groups of genus g covers of the projective line over C. We use this to construct, for each nonnegative integer gl a PAC field K of characteristic 0 which is g-Hilbertian but not Hilbertian.