Mumford's formula on the universal Picard stack

TL;DR

This paper constructs a derived pushforward of the r-th root of the universal line bundle over the Picard stack of genus g prestable curves, providing new formulas and applications to conjectures in algebraic geometry.

Contribution

It introduces a new derived pushforward construction and formulas for the universal Picard stack, connecting to Mumford's formula and conjectures on Chiodo classes.

Findings

Derived pushforward formula in terms of tautological generators

Recovery of Mumford's and related formulas via pullback

Proof of a conjecture relating higher powers of r to the double ramification cycle

Abstract

We construct a derived pushforward of the r-th root of the universal line bundle over the Picard stack of genus g prestable curves carrying a line bundle. We prove a number of basic properties, and give a formula in terms of standard tautological generators. After pullback, our formula recovers formulae of Mumford, of the first-named author, and of Pagani--Ricolfi--van Zelm. We apply these constructions to prove a conjecture expressing the coefficients of higher powers of r in the so-called `Chiodo classes' to the double ramification cycle, and to give a formula for the r-spin logarithmic double ramification cycle.