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，从密码学到​​计算机科学，许多科学和工程领域的重要系统都可以用多项式方程组来描述。代数几何研究此类多项式系统的解。通常，这些系统的解根据这些系统的定义方程中的参数以连续方式变化，从而导致解的参数化空间。一个难以研究的系统可能与一个易于求解的系统位于相同的参数空间中，从而允许人们推断出困难系统的几何约束。该项目研究数学和物理学中普遍存在的一些参数空间的几何形状，即所谓的向量丛模空间和曲线模空间。该研究通过将这些空间与更简单的空间联系起来来计算它们的几何不变量。最近一项名为布里奇兰稳定性的突破使得许多以前无法计算的不变量得以计算。该研究项目旨在扩展这些成果。 此外，该项目还通过研究生和博士后参与研究，有助于培养下一代数学研究人员。布里奇兰稳定性条件为研究滑轮模量空间的几何形状提供了新的工具。该项目旨在将布里奇兰稳定性条件与双有理几何的最新进展相结合，研究有关表面滑轮模空间和曲线模空间几何的长期存在的问题。这些模空间在数学的许多分支中发挥着核心作用。因此，理解它们的上同调和双有理几何至关重要。该项目涉及三个部分。首先，在与合作者合作的基础上，PI 将研究表面滑轮模空间的双有理几何。他将计算它们的双有理不变量，例如充足、可移动和有效除数的锥体，重点关注 Hirzebruch 曲面、del Pezzo 曲面和某些一般类型的曲面。其次，PI 将根据其他特殊集合来研究射影空间中理想的点和曲线组的分辨率。就平面而言，如果根据适当选择的特殊集合来解析滑轮，则几何形状会显着简化；这项工作将把相同的想法应用于高维射影空间上的滑轮。第三，PI将继续开展关于曲线模空间上更高余维循环的有效锥、Kawamata-Morrison圆锥猜想以及均匀空间中循环的极值性质的合作项目。