# Equivariant birational maps and resolutions of categorical quotients

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Dan Edidin, University of Missouri
Fine Hall 322

If $X^{ss}$ is the set of semi-stable points for a linearized action of a reductive group on a smooth projective variety $X$ then there two procedures (Kirwan's procedure or change of linearization) for constructing a partial resolution of singularities of the categorical quotient $X^{ss}/G$. Both involve finding an equivariant birational map $\tilde{X} \to X^{ss}$ with $\tilde{X}$ smooth such that $G$ acts properly on $\tilde{X}$ and the induced map on quotients is proper and birational. A natural question to ask is whether (and to what extent) this procedure can be replicated for non-GIT quotients. We consider the problem for actions of diagonalizable groups and show that there is a simple combinatorial procedure that replicates Kirwan's construction for non-projective toric varieties. This talk is based on joint work with Yogesh More.