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，都是用多项式方程系统来建模的。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族中变化。族中的一些方程更容易解，更复杂系统的性质可以从较简单系统的解中推导出来。研究人员研究由多项式方程定义的某些空间的几何，这些空间在数学和物理学中无处不在，称为向量束的模空间。他们计算这些空间的几何不变量，通过将它们与更简单的空间联系起来，使用最近的突破，称为布里奇兰稳定性。研究人员还致力于培养下一代美国科学家和研究人员。在这个项目中，他们将训练本科生、研究生和博士后研究人员使用布里奇兰稳定的新技术。重点研究小组拨款将支持这些年轻研究人员访问和与几位资深研究人员合作，并参加有关该主题的会议和研讨会。调查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地区。向量束的模空间是代数几何中的基本对象，在交换代数、表示理论、组合学和数学物理中都有应用。在过去的五年中，布里奇兰稳定性条件已经彻底改变了对曲面上矢量束的模空间的理解。它们允许在这些模空间上计算充足和有效的除数锥，并导致解决长期存在的问题，如某些K3型hyperkähler流形上的拉格朗日纤振的存在性和平面上一般滑轮的高秩插值问题。将这些新技术应用于曲面和三折矢量束模空间几何中的中心问题是及时的。本课题主要围绕以下三个方面展开：(1)利用bridgeeland稳定性证明上同调消失结果，从而构造曲面上的Ulrich束和曲面上向量束模空间上的三折有效Brill-Noether除数。给出勒波蒂埃奇异对偶猜想的应用。(2)确定曲面和三折上的特殊束（如Lazarsfeld-Mukai束或零相关束）何时为桥地稳定。将稳定性应用于经典的协同和Koszul上同调问题。(3)研究桥稳物体过壁模空间的双几何特性。研究人员计划通过参与该项目的研究，培养10名本科生、10名研究生和7名博士后。

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.