The Picard rank conjecture for the Hurwitz spaces of degree up to five
  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. 
Anand Deopurkar and Anand Patel
math.AG