Euler characteristics of the universal Picard stack
Siddarth Kannan
TL;DR
The paper computes S_n-equivariant, weight-graded and topological Euler characteristics for the universal Picard stack Pic_{g,n}^d over the moduli space M_{g,n}. It introduces generating functions 𝔍_g^k and 𝔃_g^k and proves a simple, explicit transform 𝒯 on symmetric-function generators that relates the Picard data to the moduli data, with a key limit lim_{x→1}(1−x)𝒯(𝔃_g^k) = ∑_{i=0}^{⌊k/2⌋} 𝔍_g^{k−2i}. This yields closed formulas for 𝔍_g^0 from the Chan–Faber–Galatius–Payne (CFGP) weight-zero formula and for topological invariants via Gorsky’s formula, as well as a closed formula for χ_{top}(Pic_g). The approach combines a geometric reduction via Poincaré line bundles and relative symmetric powers with a robust symmetric-function framework (plethysm, Δ, and the transform 𝒯), connecting to Hassett’s weighted-curves and graph-complex perspectives. The results illuminate the structure of Euler characteristics under abelian fibrations and provide computable, representation-theoretic expressions that interface with tropical and moduli-theoretic viewpoints, offering a compact, algebraic route to understanding Picard-stack invariants.
Abstract
We study $\mathbb{S}_n$-equivariant topological and weight-graded compactly-supported 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 a corollary, we also obtain a closed formula for the topological Euler characteristic of $\mathrm{Pic}^d_g$. Our weight-zero calculation is a corollary of a general result passing from the weight-graded Euler characteristics of $\mathcal{M}_{g, n}$ to those of $\mathrm{Pic}_{g,n}^d$.