Pixton's formula and Abel-Jacobi theory on the Picard stack
Younghan Bae, David Holmes, Rahul Pandharipande, Johannes Schmitt, Rosa Schwarz
备注：
122 pages. v2: This is the Author Accepted Manuscript, to appear in Acta Mathematica (open access). Main change since v1 is addition of Section 6.2 discussing a refinement of the definition (due to Marcus and Wise) of logarithmic rubber maps. Comments still very welcome
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.
For $θ$ a small generic universal stability condition of degree $0$ and $A$ a vector of integers adding up to $k(2g-2)$, the spaces $\overline{M}_{g,n}^θ$ resolving the Abel-Jacobi section to the compactified Jacobian Pic^θconstructed in the work of Abreu-Pacini and Holmes-Molcho-Pandharipande-Pixton-Schmitt are observed to lie inside the space $\textbf{Div}$ of Marcus and Wise, and their pullback to the rubber space of loc. cit to be smooth. This provides smooth and modular blowups $\widetilde{M}_{g,n}^θ$ of the moduli space of stable curves on which the logarithmic double ramification cycle …
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