Cox rings of quiver varieties

箭袋品种的考克斯环

• 批准号: 2439936
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2439936
• 关键词: Cox; rings; quiver; varieties

The research project aims to calculate generators for the Cox ring of quiver flag varieties, with a view tounderstanding degenerations of quiver flag varieties and applications in mirror symmetry.Quiver flag varieties provide a special class of varieties that generalise flag varieties of type A. Given the choice of anacyclic quiver with a unique source and a dimension vector, a natural GIT construction produces a quiver flag variety.Every such variety is a smooth Mori Dream Space that can be obtained as an iterative tower of Grassmann bundlesover a point. They provide natural ambient spaces in algebraic geometry and were used recently by Kalashnikov toproduce many new examples of Fano fourfolds.The birational geometry of a quiver flag variety is encoded in its Cox ring. This ring is known to be finitely generated,but an explicit set of generators is not known. The idea for the project is to use the fact that the Cox ring of a quiverflag variety can be interpreted as the semi-invariant ring of the corresponding quiver and dimension vector. As such,it is known that the set of Schofield semi-invariant functions provides a spanning set for the ring. It is therefore naturalto ask for an efficient collection of Scofield semi-invariants that provide a (minimal) set of algebra generators. Thespecial case where the quiver flag variety is the Grassmannian is well known, and indeed, the description of thegenerators is known as the `First Fundamental theorem of invariant theory'.Going deeper, if generators can be understood, then there are two natural questions: first, how to compute relations,thereby generalising the `Second Fundamental Theorem of Invariant Theory'; and second, how to use thesegenerators to compute (toric) degenerations of the quiver flag variety. The programme of Kalashnikov then allowsone to compute explicit mirror partners to her four-dimensional Fano varieties obtained as zero-loci in quiver flagvarieties.

该研究项目旨在计算箭旗品种的考克斯环的生成元，以期了解箭旗品种的退化及其在镜像对称中的应用。箭旗品种提供了一类特殊的品种，它概括了A型旗品种。给定一个具有唯一源和维向量的非循环向量的选择，自然的GIT构造产生一个非循环向量簇，每个这样的簇都是一个光滑的Mori Dream Space，可以作为一个点上的格拉斯曼向量的迭代塔来获得。它们在代数几何中提供了自然的环境空间，最近被卡拉什尼科夫用来产生许多新的Fano四重的例子。已知这个环是双生成的，但不知道一组明确的生成元。这个项目的想法是利用这样一个事实，即一个quiverflag簇的考克斯环可以被解释为相应的维数和维数向量的半不变环。因此，已知斯科菲尔德半不变函数的集合提供了环的生成集。因此，很自然地要求一个有效的集合斯科菲尔德半不变量，提供了一个（最小）集的代数生成器。我们已经知道了Grassmannian是一个特殊的情形，而且对生成元的描述也被称为“不变论第一基本定理”。更深入地说，如果生成元可以被理解，那么就有两个自然的问题：第一，如何计算关系，从而推广了“不变论第二基本定理”;第二，如何使用这些生成器来计算（复曲面）退化的国旗品种。卡拉什尼科夫的程序，然后allowsone计算明确的镜像合作伙伴，她的四维法诺品种获得零位点的flagvarieties。