# Reconstructing certain quiver flag varieties from a tilting bundle

• James Green

Student thesis: Doctoral ThesisPhD

### Abstract

Given a quiver flag variety Y equipped with a tilting bundle E, a construction ofCraw, Ito and Karmazyn [CIK18] produces a closed immersion f_E : Y -> M(E), where M(E) is the fine moduli space of cyclic modules over the algebra End(E).In this thesis we present two classes of examples where f_E is an isomorphism. Firstly, when Y is toric and E is the tilting bundle from [Cra11]; this generalises the well-known fact that P^n can be recovered from the endomorphism algebra of \oplus_{0\leq i \leq n} O_{P^n}(i). Secondly, when Y = Gr(n, 2), the Grassmannian of 2-dimensional quotients of k^n and E is the tilting bundle from [Kap84]. In each case, we give a presentation of the tilting algebra A = End(E) by constructing a quiver Q' such that there is a surjective k-algebra homomorphism \Phi: kQ' -> A, and then give an explicit description of the kernel.
Date of Award 22 Nov 2018 English University of Bath Alastair Craw (Supervisor) & David Calderbank (Supervisor)

### Keywords

• quiver flag variety
• tilting bundle

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