Cox rings of quiver varieties

本课题旨在计算颤振旗品种的Cox环产生子，以了解颤振旗品种的退化及其在镜像对称中的应用。颤抖旗品种提供了一种特殊的品种，它概括了a型旗品种。给定具有独特来源和维度向量的无环颤振的选择，自然的GIT结构产生了颤抖旗品种。每一个这样的变化都是一个平滑的Mori梦想空间，可以作为一个迭代的Grassmann束在一个点上的塔。它们在代数几何中提供了自然的环境空间，卡拉什尼科夫最近使用它们来产生许多新的法诺四倍的例子。颤栗旗品种的双民族几何是编码在它的考克斯环。已知这个环是有限生成的，但不知道一组显式的生成器。这个项目的想法是利用这样一个事实，即一个颤振旗的Cox环可以被解释为对应的颤振和维向量的半不变环。因此，已知Schofield半不变函数集为环提供了一个生成集。因此，要求提供（最小）代数生成器集的Scofield半不变量的有效集合是很自然的。颤振旗变异体是格拉斯曼变异体的特殊情况是众所周知的，事实上，对产生子的描述被称为“不变理论的第一基本定理”。再深入一点，如果生成器可以被理解，那么自然会有两个问题：首先，如何计算关系，从而推广“不变量理论第二基本定理”；其次，如何使用这些生成器来计算颤振旗变化的（环向）退化。然后，卡拉什尼科夫的程序允许人们计算她的四维法诺变种的显式镜像伙伴，这些变种在颤枪变体中作为零位点得到。

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.