The geometry and combinatorics of compactified universal Jacobians

紧化通用雅可比行列式的几何和组合数学

• 批准号: 2271921
• 金额: --
• 依托单位: University of Liverpool
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2019
• 资助国家: 英国
• 起止时间: 2019 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2271921
• 关键词: geometry; combinatorics; compactified; universal; Jacobians

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.

我们将研究通用雅可比行列式的紧化几何，即参数化对 (C,L) 的模空间，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。曲线模空间的经典模紧化是根据稳定曲线给出的。有几种不同的模紧化允许映射到稳定曲线的模空间。 Kass 和 Pagani 最近在一系列作品中研究了所有此类紧化模同构的集合。我们将解决两个自然（且独立）的问题。1）Migliorini-Shende-Viviani 最近的工作表明，同一稳定曲线的两个紧致雅可比行列式的上同调与向量空间同构。在这里我们想问对于两个紧致的通用雅可比行列式（在相同的模空间上）是否也是如此。该项目首先需要了解 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定的了解。2) 对于一个固定的属，存在多少个非同构紧雅可比行列式？通过卡斯-帕加尼的工作，这个问题可以重新表述为计算真实环面上某个超平面排列的室数的组合问题，以某个群的作用为模。计算向量空间中超平面排列的腔室的理论是经典的，由扎夫拉斯基提出，最近的结果将该理论扩展到环面排列的情况。