论文标题
PIXTON的配方和Abel-Jacobi理论
Pixton's formula and Abel-Jacobi theory on the Picard stack
论文作者
论文摘要
令$ a =（a_1，\ ldots，a_n）$为$ d = \ sum_ {i = 1}^n a_i $的整数的向量。通过经典的Abel-Jacobi地图的部分分辨率，我们构建了一个通用的扭曲双重冲击周期$ \ Mathsf {dr}^{\ Mathsf {op}} _ {g，a} $作为PICARD stack stack $ \ Mathfrak { $ d $ line Bundle。施工方法遵循日志（和B式）方法，即标准的双重分支周期，在曲线的模量空间上具有典型的曲折[arxiv：1707.02261，arxiv：1711.10341，arxiv：1708.04471]。 我们的主要结果是计算$ \ mathsf {dr}^{\ Mathsf {op}} _ {g，a} $在PICARD stack $ \ Mathfrak {pic} _ {p} _ {g，n，d} $中，通过对Pixton在重训式学环中的pixton公式的适当解释。证明中使用的基本新工具是目标品种的双重影响周期的理论[ARXIV：1812.10136]。 PICARD堆栈上的公式是从[arxiv：1812.10136]的目标品种获得的$ \ mathbb {cp}^n $中的限制$ n \ rightarrow \ rightarrow \ infty $。结果可以看作是亚伯 - 雅各比理论中的普遍计算。 由于$ \ mathsf {dr}^{\ mathsf {op}} _ {g，a} $的计算，在Picard stack $ \ mathfrak {pic} _ {p} _ {g，n，d} $上$ \ OVILLINE {\ MATHCAL {M}} _ {G，N} $由Pixton的公式完全给出（如[Arxiv：1508.07940]的附录中，并在[Arxiv：Arxiv：1607.08429]中给出。 [Arxiv：1909.11981]中证明的基本类别的比较结果在我们的论点中起着至关重要的作用。我们还证明了Picard stack $ \ mathfrak {pic} _ {g，n，d} $与Pixton公式相关的重言式戒指中的一组关系。
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 $n \rightarrow \infty$. 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.
