Pixton’s formula and Abel–Jacobi theory on the Picard stack

Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel–Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{{\sf op}}_{g,A}$ as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus-$g$ curves carrying a degree $d$ line bundle. The method of construction follows the $\log$ (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [37], [38], [56].

Our main result is a calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton’s formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [42]. The formula on the Picard stack is obtained from [42] for target varieties $\mathbb{CP}^n$ in the limit $n\to \infty$. The result may be viewed as a universal calculation in Abel–Jacobi theory.

As a consequence of the calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton’s formula (as conjectured in [28, Appendix] and [72]). The comparison result of fundamental classes proven in [40] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated with Pixton’s formula.

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Received 5 May 2020

Received revised 11 May 2021

Accepted 16 August 2021

Published 18 July 2023