The Picard Group of the Stack of Pointed Smooth Cyclic Covers of the Projective Line

Abstract

We study the stack $\mathcal{H}_{r,g,n}$ of $n$-pointed smooth cyclic covers of degree $r$ between smooth curves of genus $g$ and the projective line. We give two presentations of an open substack of $\mathcal{H}_{r,g,n}$ as a quotient stack, and we study its complement. Using this, we compute the integral Picard group of $\mathcal{H}_{r,g,n}$. Moreover, we obtain a very explicit description of the generators of the Picard group, which have evident geometric meaning. As a corollary of the computation, we get the integral Picard group of the stack $\mathcal{H}_{g,n}$ of $n$-pointed hyperelliptic curves of genus $g$. Finally, taking $g=2$ and recalling that $\mathcal{H}_{2,n}=\mathcal{M}_{2,n}$, we obtain $\mathrm{Pic}(\mathcal{M}_{2,n})$.