Picard叠上的Pixton公式和Abel-Jacobi理论

IF 5.4 3区 材料科学 Q2 CHEMISTRY, PHYSICAL
Younghan Bae, D. Holmes, R. Pandharipande, Johannes Schmitt, Rosa Schwarz
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引用次数: 27

摘要

设$A=(A_1,\ldots,A_n)$是整数的向量,其中$d=\sum_{i=1}^n A_i$。通过对经典Abel-Jacobi映射的部分解析,我们在Picard堆栈$\mathfrak上构造了一个通用的双分支环$\mathsf{DR}^{\mathsf}op}}_{g,a}$作为运算Chow类{Pic}_$n$的{g,n,d}$-带次$d$线丛的尖亏格$g$曲线。构造方法遵循log(和b-Chow)方法,在曲线的模量空间上具有正则扭曲的标准双分支循环[arXiv:1707.02261,arXiv:1711.10341,arXiv:1708.04471]。我们的主要结果是在Picard堆栈$\mathfrak上计算$\mathsf{DR}^{\mathsf}op}}_{g,a}${Pic}_{g,n,d}$通过对重言环中皮克斯顿公式的适当解释。在证明中使用的基本新工具是目标品种的双分枝循环理论[arXiv:1812.10136]。Picard堆栈上的公式是从[arXiv:1812.10136]中获得的,目标品种$\mathbb{CP}^n$在极限$n\rightarrow\infty$中。这个结果可以看作是阿贝尔-雅可比理论中的一个普遍计算。作为Picard堆栈$\mathfrak上$\mathsf{DR}^{\mathsf}op}}_{g,a}$的计算结果{Pic}_{g,n,d}$,我们证明了$\overline{\mathcal{M}}_{g,n}$中扭曲亚纯微分模空间的基类是由Pixton公式(如[arXiv:150807940]附录和[arXiv:1607.08429]中所推测的)给出的。我们还证明了Picard堆栈$\mathfrak的重言环中的关系集{Pic}_{g,n,d}$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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 $\mathsf{DR}^{\mathsf{op}}_{g,A}$ 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.
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来源期刊
ACS Applied Energy Materials
ACS Applied Energy Materials Materials Science-Materials Chemistry
CiteScore
10.30
自引率
6.20%
发文量
1368
期刊介绍: ACS Applied Energy Materials is an interdisciplinary journal publishing original research covering all aspects of materials, engineering, chemistry, physics and biology relevant to energy conversion and storage. The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrate knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important energy applications.
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