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 ${\mathcal{H}}_{d,g}$ 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.

Added a figure. Fixed typos and numerical mistakes

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