Vol. 9, No. 2, 2015

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

Anand Deopurkar and Anand Patel

Vol. 9 (2015), No. 2, 459–492
Abstract

We prove that the rational Picard group of the simple Hurwitz space ${\mathsc{ℋ}}_{d,g}$ 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.

Keywords
Hurwitz space, Picard group
Primary: 14H10
Secondary: 14C22