Cox rings of quiver varieties

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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.