The geometry and combinatorics of compactified universal Jacobians

我们将研究通用雅可比矩阵的紧化几何，即参数化对（C，L）的模空间，其中C是给定属的光滑代数曲线，L是C上某固定度的线束。有几种不同的模紧化可以映射到稳定曲线的模空间。所有这些紧化模同构的集合最近已经被Kass和Pagani在一系列的工作中研究。我们将解决两个自然的（独立的）问题。1) Migliorini-Shende-Viviani最近的工作证明了同一稳定曲线的两个紧化雅可比矩阵的上同构在向量空间中是同构的。这里我们想问的是，对于两个紧化的全称雅可比矩阵（在相同模空间上）是否同样成立。这个项目首先需要了解Migliorini-Shende-Viviani结果证明的细节，这需要对交上同调和Beilinson-Bernstein-Deligne的经典分解定理有一些了解。2)对于一个固定的格存在多少个非同构紧化雅可比矩阵？通过Kass-Pagani的工作，这个问题可以被重新表述为计算实环面上某超平面排列的室数的组合问题，模取某群的作用。计算向量空间中超平面排列的腔室的理论是经典的，是由Zavlasky提出的，最近的结果将该理论扩展到环面排列的情况。

We will study the geometry of the compactification of the universal Jacobian, the moduli space that parametrises pairs (C,L) where C is a smooth algebraic curve of given genus and L is a line bundle of some fixed degree on C. A classical modular compactification of the moduli space of curves is given in terms of stable curves. There are several different modular compactifications that admit a map to the moduli space of stable curves. The set of all such compactifications modulo isomorphisms has been recently studied in a series of works by Kass and Pagani. We will address two natural (and independent) questions.1) Recent work by Migliorini-Shende-Viviani shows that the cohomology of two compactified Jacobians of the same stable curve are isomorphic as vector spaces. Here we want to ask if the same is true for two compactified universal Jacobians (over the same moduli space). This project will first require to understand the details of the proof of Migliorini-Shende-Viviani's result, which requires developing some understanding of intersection cohomology and of the classical decomposition theorem by Beilinson-Bernstein-Deligne.2) How many non-isomorphic compactified Jacobians exist for a fixed genus? By the work of Kass-Pagani this question can be reformulated as the combinatorial problem of counting the number of chambers of a certain hyperplane arrangement on a real torus, modulo the action of a certain group. The theory to count the chambers of a hyperplane arrangement in a vector space is classical and due to Zavlasky, and recent results extend that theory to the case of arrangements on a torus.