Euler characteristics of the universal Picard stack

We study ${\mathbb{S}}_n$-equivariant weight-graded and topological Euler characteristics of the universal Picard stack ${\mathrm{P}\mathrm{i}\mathrm{c}}_{g,n}^d\to {\mathcal{M}}_{g,n}$ of degree-$d$ line bundles over ${\mathcal{M}}_{g,n}$. We prove that in the weight-zero and topological cases, the generating function for Euler characteristics of ${\mathrm{P}\mathrm{i}\mathrm{c}}_{g,n}^d$ is obtained from the corresponding one for ${\mathcal{M}}_{g,n}$ by an extremely simple combinatorial transformation. This lets us deduce closed formulas for the two generating functions, taking as input the Chan–Faber–Galatius–Payne formula in the weight-zero case and Gorsky’s formula in the topological case. As an immediate corollary, we obtain closed formulas for the weight-zero and topological Euler characteristics of ${\mathrm{P}\mathrm{i}\mathrm{c}}_g^d$. Our weight-zero calculations follow from a general result passing from the weight-graded Euler characteristics of ${\mathcal{M}}_{g,n}$ to those of ${\mathrm{P}\mathrm{i}\mathrm{c}}_{g,n}^d$.