The geometry and combinatorics of compactified universal Jacobians

我们将研究万有雅可比矩阵的紧化的几何学，万有雅可比矩阵是一个模空间，它使对（C，L）参数化，其中C是给定亏格的光滑代数曲线，L是C上某个固定次数的线丛。利用稳定曲线给出了曲线模空间的一个经典模紧化。有几种不同的模紧化允许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在卡斯和帕加尼的一系列著作中得到了研究。我们将解决两个自然的（和独立的）问题。1）Migliorini-Shende-Viviani最近的工作表明，同一稳定曲线的两个紧化Jacobian的上同调是同构的向量空间。这里我们想问，对于两个紧化的泛雅可比行列式（在同一模空间上），是否也是如此。这个项目首先需要理解Migliorini-Shende-Viviani结果的证明细节，这需要对Beilinson-Bernstein-Deligne的交上同调和经典分解定理有一定的理解。2）对于一个固定的亏格，存在多少个非同构的紧化Jacobian？通过卡斯-帕加尼的工作，这个问题可以被重新表述为计算真实的环面上某个超平面排列的腔室数量的组合问题，以某个组的作用为模。理论计算室的超平面安排在一个向量空间是经典的，由于Zavlasky，最近的结果扩展了该理论的情况下安排的环面。

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.