The problem of computing invariants of natural stacks of curves has a long history, starting from Mumford's seminal paper on the Picard group of the stack of 1-pointed elliptic curves. The Picard group of the stack \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth curves of genus \(g\geq3\) was later computed over \(\mathbb{C}\) by Harer.

We study the closed substack \(\mathcal{H}_{g,n}\) in \(\mathcal{M}_{g,n}\) of \(n\)-pointed smooth hyperelliptic curves of genus \(g\), and compute its Picard group. As a corollary, taking \(g=2\) and recalling that \(\mathcal{H}_{2,n}=\mathcal{M}_{2,n}\), we obtain \(\mathrm{Pic}(\mathcal{M}_{2,n})\) for all \(n\).

Moreover, we give a very explicit description of the generators of the Picard group, which have evident geometric meaning.