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，都是由多项式方程组建模的。代数几何研究这类系统的解。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家族中变化。族中的一些方程更容易求解，更复杂系统的性质可以从简单系统的解推导出来。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形状，称为向量丛的模空间。他们计算这些空间的几何不变量，通过使用最近的突破称为Bridgeland稳定性将它们与更简单的空间联系起来。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Bridgeland稳定性的新技术。重点研究小组赠款将支持这些年轻的研究人员访问并与几位高级研究人员合作，并参加有关该主题的会议和研讨会。研究人员还将组织两个大型会议和四个研讨会，以帮助吸引年轻人才到该领域。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。在过去的五年中，Bridgeland稳定性条件已经彻底改变了对曲面上向量丛模空间的理解。他们允许计算的充分和有效的锥的因子对这些模空间，并导致解决长期存在的问题，如存在的拉格朗日纤维化某些hyperkähler流形的K3型和高阶插值问题的一般层的平面上。这是及时的应用这些新技术的中心问题，在几何模空间的向量丛的表面和三重。本专题研究组的研究主要集中在三个方面：（1）利用Bridgeland稳定性证明上同调消失结果，从而构造曲面上的Ulrich丛和曲面上向量丛模空间上的三重有效Brill-Noether因子。Le Potier的奇异对偶猜想(2)确定特殊的丛，如Lazarssen-Mukai丛或曲面上的零相关丛和三重丛，何时是Bridgeland稳定的。将稳定性应用于合冲和Koszul上同调的经典问题。(3)研究Bridgeland稳定物体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.