Picard Groups of Hurwitz Schemes

Fix positive integers g and d, and let k be an algebraically closed field of characteristic 0. Consider the moduli space Hd,g of d-fold simply branched covers C P1 , where C is a smooth curve of genus g. Harris and Mumford discovered a modular compactification, H d,g , obtained by allowing "admissible covers", where both domain and target curves acquire nodes. We will examine the rational Picard group of H d,g , which is conjectured to be generated by boundary classes.