# Pixton's formula and Abel-Jacobi theory on the Picard stack

Younghan Bae, David Holmes, Rahul Pandharipande, Johannes Schmitt, Rosa Schwarz

Younghan Bae, David Holmes, Rahul Pandharipande, Johannes Schmitt, Rosa Schwarz

**18/4/20**Published in : arXiv:2004.08676

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}^{\mathsf{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 [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471].

Our main result is a calculation of

\mathsf{DR}^{\mathsf{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 [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] for target varieties

\mathbb{CP}^n |

in the limit

n |

. The result may be viewed as a universal calculation in Abel-Jacobi theory.

As a consequence of the calculation of

\mathsf{DR}^{\mathsf{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 the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] 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 to Pixton's formula.