Cox rings of quiver varieties

该研究项目旨在计算箭图旗簇的Cox环的生成元，以了解箭图旗簇的退化及其在镜像对称中的应用.箭图旗簇提供了一类特殊的簇，它推广了A类旗簇.给定具有唯一源和维度向量的非循环箭图，自然GIT构造产生箭旗簇.每个箭图簇都是光滑的Mori梦想空间，可以作为Grassmann分支在一点上的迭代塔来获得.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四重数的例子。箭图旗簇的双调几何被编码在它的Cox环中。这个环已知是有限生成的，但显式生成元集是未知的。这个项目的想法是利用这样一个事实，即箭旗变种的COX环可以解释为对应的箭图和维矢量的半不变环。因此，众所周知，Schofield半不变函数集为环提供了一个支撑集。因此，要求一个有效的Scofield半不变量集合来提供一组(最小的)代数生成器是很自然的。箭旗簇是Grassman簇的特殊情况是众所周知的，实际上，对箭旗簇的描述被称为‘第一不变论基本定理’。深入地说，如果生成元能够被理解，那么有两个自然问题：第一，如何计算关系，从而推广了‘第二不变量理论基本定理’；第二，如何使用生成元来计算箭旗簇的(环状)退化。然后，Kalashnikov的程序允许Sone计算她的四维Fano品种的显式镜像伙伴，在箭形旗帜品种中获得的是零轨迹。

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.