Birational Geometry of Moduli Spaces and Bridgeland Stability

在科学和工程的许多领域中，从进化生物学到物理学，从密码学到计算机科学，重要的系统都可以用多项式方程组来描述。代数几何学研究这样的多项式系统的解。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的方式变化，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容易求解的系统处于同一个参数空间中，从而可以推导出困难系统的几何约束。本项目研究在数学和物理中普遍存在的一些参数空间的几何，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相关联来计算这些空间的几何不变量。最近的一个突破，称为Bridgeland稳定性，允许计算许多以前无法访问的不变量。本研究项目旨在推广这些成果。 此外，该项目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。Bridgeland稳定性条件为研究层的模空间几何提供了一个新的工具。这个项目旨在将联合收割机Bridgeland稳定性条件与双有理几何的最新进展相结合，以研究关于曲面上层的模空间和曲线的模空间的几何学的长期问题。这些模空间在数学的许多分支中起着核心作用。因此，理解它们的上同调和双有理几何是至关重要的。该项目包括三个部分。首先，在与合作者的工作基础上，PI将研究曲面上层的模空间的双有理几何。他将计算他们的双有理不变量，如圆锥体的充足，可移动，有效的除数，集中在Hirzebruch表面，德尔佩佐表面，和某些表面的一般类型。第二，PI将研究射影空间中点和曲线的理想层在其他例外集合中的分解。在平面的情况下，如果人们根据一个适当选择的例外集合来解析层，那么几何学就大大简化了;这项工作将把同样的思想应用于高维射影空间上的层。第三，PI将继续在曲线模空间上的高余维循环的有效锥，Kawamata-Morrison锥猜想和齐次空间中循环的极值性质方面的合作项目。

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.