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

生活中的许多过程，从信用卡交易到向日葵的生长，都是由多项式方程系统来模拟的。代数几何研究此类系统的解。这些系统的一个主要特点是，它们通过改变多项式的系数而在族中变化。这个族中的一些方程更容易求解，而更复杂的系统的性质可以从更简单的系统的解中推导出来。研究人员研究由数学和物理中普遍存在的多项式方程定义的某些空间的几何，称为向量丛的模空间。他们通过将这些空间与更简单的空间联系起来，利用最近的一项名为布里奇兰稳定性的突破来计算这些空间的几何不变量。调查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这一新技术。重点研究小组的赠款将支持这些年轻研究人员访问几名资深研究人员并与其合作，并参加关于这一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到该领域来。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。在过去的五年中，Bridgeland稳定性条件彻底改变了人们对曲面上向量丛的模空间的理解。它们允许计算这些模空间上因子的充分和有效锥，并导致解决长期存在的问题，如某些K3型超Kähler流形上的Lagrangian纤维的存在性和平面上一般层的高阶插值问题。现在正是将这些新技术应用于曲面和三折向量丛的模空间几何的中心问题的时候了。(1)利用Bridgeland稳定性证明上同调零结果，从而构造曲面上的Ulrich丛和曲面上向量丛的模空间上的三重和有效的Brill-Noether因子。给出了Le Potier的奇异对偶猜想的应用。(2)确定了特殊丛，如曲面和三重上的Lazarsfeld-Mukai丛或零相关丛何时Bridgeland稳定。将稳定性应用于合调和Koszul上同调的经典问题。(3)通过跨墙方法研究Bridgeland稳定对象的模空间的双调几何。研究人员计划通过参与该项目的研究，培训10名本科生、10名研究生和7名博士后助理。

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.