Euler characteristics of the universal Picard stack

We study $\mathbb{S}n$-equivariant weight-graded and topological Euler characteristics of the universal Picard stack $\mathrm{Pic}{g, n}^d \to \mathcal{M}{g, n}$ 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 $\mathrm{Pic}{g, n}^d$ 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 $\mathrm{Pic}^d_g$. Our weight-zero calculations follow from a general result passing from the weight-graded Euler characteristics of $\mathcal{M}{g, n}$ to those of $\mathrm{Pic}{g,n}^d$.

💡 Research Summary

The paper investigates the (S_n)-equivariant weight‑graded and topological Euler characteristics of the universal Picard stack (\mathrm{Pic}^{d}{g,n}\to\mathcal M{g,n}), which parametrises degree‑(d) line bundles on smooth (n)-pointed curves of genus (g). The main achievement is a remarkably simple combinatorial transformation that converts the known generating functions for the Euler characteristics of the moduli space of curves (\mathcal M_{g,n}) into the corresponding generating functions for the Picard stack.

Key constructions.
For (m>2g-2) the authors consider the Poincaré line bundle (\mathcal L) on the product (\mathrm{Pic}^{m}{g,n}\times{\mathcal M_{g,n}}\mathcal C_{g,n}) (where (\mathcal C_{g,n}) is the universal curve). The push‑forward (\pi_\mathcal L) is a vector bundle, and its projectivisation (S_{m,g,n}:=\mathbb P(\pi_\mathcal L)) is a smooth Deligne–Mumford stack. Using the projective bundle formula for DM stacks (Cho12) they obtain, for any weight‑grade (k), an explicit relation between the graded compactly‑supported cohomology of (S_{m,g,n}) and that of (\mathrm{Pic}_{g,n}): \