Equivariant Picard groups of the moduli spaces of some finite abelian covers of the Riemann sphere (Q2683772)

An explicit computation of the equivariant Picard group \(\mathrm{Pic}(\mathcal{M}_{g}^{A})\) is given in this note, for the moduli space of genus \(g\) curves with a group of automorphisms acting topologically like a fixed finite abelian group \(A\) of rank at least two. The computation is explicit and implies that such Picard groups are finite, in particular relating to the order of the group \(A\) and the number of branch points of a regular branched covering of the 2-sphere by a genus \(g\) curve. The author demonstrates how one can employ results of \textit{K. Kordek} [Int. Math. Res. Not. 2020, No. 23, 9293--9335 (2020; Zbl 1465.14029)], thus providing that \(\mathrm{Pic}(\mathcal{M}_{g}^{A})\) is isomorphic to the abelianization of the semidirect product \[ \mathrm{Mod}(\Sigma_{0,n}) \ltimes A, \] for \(\mathrm{Mod}(\Sigma_{0,n})\), the mapping class group of a genus 0 orientable surface with \(n\) punctures, where \(n\) is the number of branch points described above. The problem then reduces to the study of the action of \(\mathrm{Mod}(\Sigma_{0,n})\) on the group \(A\).