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.

中文翻译：

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皮卡德栈上的皮克斯顿公式和阿贝尔-雅可比理论

设$A=(a_1,\ldots,a_n)$为$d=\sum_{i=1}^n a_i$的整数向量。通过经典阿贝尔-雅可比映射的部分解析，我们构造了一个通用扭曲双分支循环 $\mathsf{DR}^{{\sf op}}_{g,A}$ 作为 Picard 堆栈 $\mathfrak{Pic}_{g,n,d}$ 上的操作 Chow 类，其中包含 $n$ 指向的 genus-$g$ 曲线，带有 $d$ 度线束。构造方法遵循 $\log$（和 b-Chow）方法来实现标准双分支循环，并在曲线的模空间上进行规范扭曲 [ 37 ]、[ 38 ]、[ 56]]。我们的主要结果是通过适当的方法在 Picard 堆栈 $\mathfrak{Pic}_{g,n,d}$ 上计算 $\mathsf{DR}^{{\sf op}}_{g,A}$同义反复环中皮克斯顿公式的解释。证明中使用的基本新工具是目标品种的双分支循环理论[ 42 ]。皮卡德栈上的公式由[ 42] 对于目标品种 $\mathbb{CP}^n$ 在限制 $n\to \infty$ 内。该结果可以被视为阿贝尔-雅可比理论中的通用计算。通过在 Picard 栈 $\mathfrak{Pic}_{g,n,d}$ 上计算 $\mathsf{DR}^{{\sf op}}_{g,A}$ 的结果，我们证明了 $\overline{\mathcal{M}}_{g,n}$ 中扭曲亚纯微分的模空间的基本类完全由 Pixton 公式给出（如 [ 28 ，附录 中的推测） ]和[ 72 ] ）。[ 40 ]中证明的基本类的比较结果在我们的论证中起着至关重要的作用。我们还证明了与 Pixton 公式相关的皮卡德堆栈 $\mathfrak{Pic}_{g,n,d}$ 的同义反复环中的关系集。