The Picard rank conjecture for the Hurwitz spaces of degree up to five

Abstract

We prove that the rational Picard group of the simple Hurwitz space ${\mathcal{\mathscr{H}}}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.