Birational Geometry of Moduli Spaces and Bridgeland Stability

模空间双有理几何与 Bridgeland 稳定性

• 批准号: 1500031
• 负责人: Izzet Coskun
• 金额: $ 18万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-15 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500031&HistoricalAwards=false
• 关键词: Birational; Geometry; Moduli; Spaces; Bridgeland

Important systems in many areas of science and engineering, ranging from evolutionary biology to physics and from cryptography to computer science, can be described by systems of polynomial equations. Algebraic geometry studies solutions of such polynomial systems. Often, the solutions of these systems vary in a continuous manner depending on parameters in the defining equations of these systems, leading to parametrized spaces of solutions. A difficult-to-study system may lie in the same parameter space as a system that is easy to solve, allowing one to deduce geometric constraints on the difficult system. This project studies the geometry of some parameter spaces that are ubiquitous in mathematics and physics, so-called moduli spaces of vector bundles and moduli spaces of curves. The research computes geometric invariants of these spaces by relating them to simpler spaces. A recent breakthrough called Bridgeland stability has allowed computation of many of the invariants that were previously inaccessible. This research project aims to extend these results. In addition, the project contributes to training the next generation of researchers in mathematics through the involvement of graduate students and postdoctoral associates in the research. Bridgeland stability conditions provide a new tool for studying the geometry of moduli spaces of sheaves. This project aims to combine Bridgeland stability conditions with recent advances in birational geometry to study longstanding questions about the geometry of moduli spaces of sheaves on surfaces and moduli spaces of curves. These moduli spaces play a central role in many branches of mathematics. Consequently, it is crucial to understand their cohomology and birational geometry. The project involves three parts. First, building on work with collaborators, the PI will study the birational geometry of the moduli spaces of sheaves on surfaces. He will compute their birational invariants such as the cones of ample, movable, and effective divisors, concentrating on Hirzebruch surfaces, del Pezzo surfaces, and certain surfaces of general type. Second, the PI will study resolutions of ideal sheaves of points and curves in projective space in terms of other exceptional collections. In the case of the plane, the geometry is significantly simplified if one resolves sheaves in terms of an appropriately chosen exceptional collection; this work will apply the same ideas to sheaves on higher-dimensional projective spaces. Third, the PI will continue collaborative projects on the effective cones of higher codimension cycles on moduli spaces of curves, the Kawamata-Morrison Cone Conjecture, and extremality properties of cycles in homogeneous spaces.

从进化生物学到物理学以及从密码学到​​计算机科学的许多领域的重要系统，可以通过多项式方程式来描述。代数几何研究此类多项式系统的解决方案。通常，这些系统的解决方案会以连续的方式变化，具体取决于这些系统定义方程式中的参数，从而导致解决方案的参数空间。一个难以研究的系统可能与易于求解的系统相同的参数空间，从而允许人们在困难系统上推断几何约束。该项目研究了一些在数学和物理学中无处不在的参数空间的几何形状，矢量束的所谓模量和曲线模量空间。该研究通过将它们与更简单的空间联系起来来计算这些空间的几何不变。最近的一个名为Bridgeland稳定性的突破允许计算以前无法访问的许多不变式。该研究项目旨在扩展这些结果。 此外，该项目通过研究生和博士后伙伴参与研究，有助于培训下一代研究人员在数学方面的培训。 Bridgeland稳定性条件提供了一种新工具，用于研究带束带的模量空间的几何形状。该项目旨在将Bridgeland稳定性条件与Birational几何形状的最新进展相结合，以研究有关曲线表面和模量曲线空间上束模束空间的几何形状的长期问题。这些模量空间在数学的许多分支中起着核心作用。因此，了解他们的同胞和异性几何学至关重要。该项目涉及三个部分。首先，在与合作者合作的基础上，PI将研究表面上带束带的模束空间的异性几何形状。他将计算出他们的异常不变性，例如充足，可移动和有效的除数的锥体，专注于内姆布鲁克表面，del pezzo表面以及一般类型的某些表面。其次，PI将根据其他特殊集合来研究投影空间中的理想束带和曲线的决议。在平面的情况下，如果一个人根据适当选择的特殊收集来解决滑轮，则几何形状会显着简化；这项工作将在高维投影空间上应用相同的想法。第三，PI将继续在曲线的模量空间，川田莫里森锥体的模量空间上的较高的编成循环的有效锥上进行协作项目。