On the Kodaira dimension of Hurwitz spaces

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.
16 pages. Final version, to appear in a special issue of Mathematische Zeitschrift in honor of Olivier Debarre