Title: Quiver moduli and effective global generation
Abstract: In this talk, I will revisit and refine a classical result of King that moduli spaces of semistable representations of acyclic quivers are projective using modern methods. For this, I will have to briefly review the theory of good and adequate moduli spaces for algebraic stacks, introduced by Alper and being developed by Alper, Halpern-Leistner, Heinloth and others. I will define the stack of semistable quiver representations and use a recent existence result to explain why it admits an adequate moduli space. Our methods allow us to improve the classical results: I will define a determinantal line bundle on the stack which descends to a semiample line bundle on the moduli space and provide effective bounds for global generation. For an acyclic quiver, we can observe that this line bundle is ample and thus the adequate moduli space is projective over an arbitrary noetherian base. This talk is based on a recent preprint with Belmans, Damiolini, Franzen, Hoskins, Tajakka (https://arxiv.org/abs/2210.00033).