FRG: Collaborative Research: Moduli Spaces, Birational Geometry, and Stability Conditions

FRG：协作研究：模空间、双有理几何和稳定性条件

• 批准号: 1663813
• 负责人: Aaron Bertram
• 金额: $ 16.29万
• 依托单位: University of Utah
• 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=1663813&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Moduli; Spaces

Many processes in life, ranging from credit card transactions to the growth of a sunflower, are modeled by systems of polynomial equations. Algebraic geometry studies solutions of such systems. A major feature of these systems is that they vary in families by varying the coefficients of the polynomials. Some equations in the family are easier to solve, and properties of more complicated systems can be deduced from the solutions of the simpler systems. The investigators study the geometry of certain spaces defined by polynomial equations that are ubiquitous in mathematics and physics, called moduli spaces of vector bundles. They compute geometric invariants of these spaces by relating them to simpler spaces using a recent breakthrough called Bridgeland stability. The investigators are also dedicated to training the next generation of U.S. scientists and researchers. In this project, they will train undergraduate, graduate, and postdoctoral researchers to use the new technique of Bridgeland stability. The Focused Research Group grant will support these young researchers to visit and collaborate with several senior researchers and to attend conferences and workshops on the topic. The investigators will also organize two large conferences and four workshops to help attract young talent to the area.Moduli spaces of vector bundles are fundamental objects in algebraic geometry, with applications to commutative algebra, representation theory, combinatorics, and mathematical physics. In the last five years, Bridgeland stability conditions have revolutionized the understanding of moduli spaces of vector bundles on surfaces. They have allowed the computation of the ample and effective cones of divisors on these moduli spaces and led to the solution of longstanding problems such as the existence of Lagrangian fibrations on certain hyperkähler manifolds of K3 type and the higher rank interpolation problem for general sheaves on the plane. It is timely to apply these new techniques to central problems in the geometry of moduli spaces of vector bundles on surfaces and threefolds. This Focused Research Group project centers on three lines of inquiry:(1) Prove cohomology vanishing results using Bridgeland stability and consequently construct Ulrich bundles on surfaces and threefolds and effective Brill-Noether divisors on moduli spaces of vector bundles on surfaces. Give applications to Le Potier's Strange Duality Conjecture.(2) Determine when special bundles, such as Lazarsfeld-Mukai bundles or null-correlation bundles on surfaces and threefolds, are Bridgeland stable. Apply the stability to classical problems on syzygies and Koszul cohomology.(3) Study the birational geometry of moduli spaces of Bridgeland stable objects via wall-crossing. The investigators plan to train ten undergraduates, ten graduate students, and seven postdoctoral associates through research involvement in the project.

生活中的许多过程，从信用卡交易到向日葵的生长，都是通过多项式方程组来建模的。代数几何研究此类系统的解决方案。这些系统的一个主要特征是它们通过改变多项式的系数而在系列中发生变化。该族中的一些方程更容易求解，并且可以从更简单系统的解中推导出更复杂系统的性质。研究人员研究由数学和物理学中普遍存在的多项式方程定义的某些空间的几何形状，称为向量丛的模空间。他们利用最近的一项名为布里奇兰稳定性的突破，将这些空间与更简单的空间联系起来，计算这些空间的几何不变量。研究人员还致力于培训下一代美国科学家和研究人员。在这个项目中，他们将培训本科生、研究生和博士后研究人员使用布里奇兰稳定性新技术。重点研究小组拨款将支持这些年轻研究人员访问几位资深研究人员并与之合作，并参加有关该主题的会议和研讨会。研究人员还将组织两次大型会议和四次研讨会，以帮助吸引年轻人才来到该领域。向量丛的模空间是代数几何中的基本对象，应用于交换代数、表示论、组合学和数学物理。在过去五年中，布里奇兰稳定性条件彻底改变了对曲面上矢量丛模空间的理解。它们允许在这些模空间上计算充足且有效的除数锥，并解决了长期存在的问题，例如 K3 类型的某些超卡勒流形上拉格朗日纤维的存在以及平面上一般滑轮的高阶插值问题。现在是将这些新技术应用于曲面和三重向量束模空间几何的中心问题是及时的。该重点研究小组项目集中于三个研究方向：（1）利用布里奇兰稳定性证明上同调消失结果，从而构造曲面上的乌尔里希丛以及曲面上向量丛模空间上的三重和有效的布里尔-诺特约数。给出勒波捷奇异对偶猜想的应用。(2) 确定特殊丛，例如曲面和三重上的拉扎斯菲尔德-穆凯丛或零相关丛，何时是布里奇兰稳定的。将稳定性应用于syzygies和Koszul上同调的经典问题。(3)通过穿墙研究Bridgeland稳定对象模空间的双有理几何。 研究人员计划通过参与该项目的研究来培养 10 名本科生、10 名研究生和 7 名博士后。