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Algebraic and Tropical Moduli Spaces and Brill-Noether Theory

代数和热带模空间以及布里尔-诺特理论

基本信息

批准号：
1701924
负责人：
Melody Chan
金额：
$ 18万
依托单位：
Brown University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1701924&HistoricalAwards=false
关键词：
Algebraic Tropical Moduli Spaces Brill

项目摘要

Algebraic geometers study spaces that are (locally) defined by polynomial equations. One powerful method of studying these spaces, which has seen exciting recent developments, is the method of degenerations. Applying this method allows to reduce complicated geometric objects to configurations of simple ones replacing some geometric aspects by combinatorial ones. Such degenerations can be useful for study of geometric problems: the very rough philosophy is that one studies the combinatorics (i.e., the discrete data) of the pieces in order to deduce things about the more complicated original space. The research supported by this award will center on using modern degeneration techniques, especially those from the field of tropical geometry, to study classical spaces from algebraic geometry. The main goal of tropical geometry is transforming questions about algebraic varieties into questions about polyhedral complexes. A process called tropicalization attaches a polyhedral complex to an algebraic variety. The polyhedral complex, a combinatorial object, encodes some of the geometry of the original algebraic variety. The first main project involves using new techniques from tropical geometry and combinatorial topology to compute top-weight rational cohomology of the moduli space of curves, finding explicit new cohomology classes therein. A complementary aspect of this project is to develop stack-theoretic foundations for tropical moduli spaces. The second project studies the geometry of Brill-Noether varieties of curves, i.e. moduli spaces of linear series on curves. The approach builds on recent advances in the moduli theory of limit linear series, and will yield a refined understanding of Brill-Noether varieties. This project will also uncover further connections between Brill-Noether theory and the combinatorics of graphs and Young tableaux. These connections are then amplified in the third main direction of research, which consists of several combinatorial investigations that will shed additional light on the close connection between graphs and algebraic curves.

代数几何学家研究由多项式方程（局部）定义的空间。研究这些空间的一个强有力的方法是简并方法，它最近取得了令人兴奋的进展。应用这种方法可以减少复杂的几何对象的配置简单的替代组合的一些几何方面。这种退化对于几何问题的研究可能是有用的：非常粗略的哲学是，人们研究组合学（即，离散数据），以便推断出更复杂的原始空间的东西。该奖项支持的研究将集中在使用现代退化技术，特别是来自热带几何领域的技术，从代数几何研究经典空间。热带几何的主要目标是将代数簇的问题转化为多面体复形的问题。一个称为热带化的过程将一个多面体复形附加到一个代数簇上。多面体复形是一个组合对象，它编码了原始代数簇的一些几何。第一个主要项目涉及使用新的技术从热带几何和组合拓扑来计算顶权重有理上同调的模空间的曲线，找到明确的新的上同调类。该项目的一个补充方面是开发热带模空间的堆栈理论基础。第二个项目研究Brill-Noether曲线簇的几何，即曲线上线性级数的模空间。该方法建立在极限线性级数的模理论的最新进展，并将产生一个精确的理解Brill-Noether品种。该项目还将揭示Brill-Noether理论与图形组合学和Young tableaux之间的进一步联系。这些连接，然后放大的第三个主要研究方向，其中包括几个组合的调查，将进一步阐明图和代数曲线之间的密切联系。