Euler characteristics of the universal Picard stack

TL;DR

This paper derives explicit formulas for the Euler characteristics of the universal Picard stack over moduli spaces of curves, revealing simple combinatorial relations and extending known formulas in algebraic geometry.

Contribution

It introduces a method to compute Euler characteristics of the universal Picard stack from those of the moduli space of curves, providing closed-form formulas in weight-zero and topological cases.

Findings

Derived closed formulas for Euler characteristics of the Picard stack

Established a simple combinatorial transformation between Euler characteristics of stacks and moduli spaces

Extended known formulas to new contexts in algebraic geometry

Abstract

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 $\mathrm{Pic}_{g,n}^d$.