Seminar Algebra and Geometry: Ivan Arzhantsev (Moscow State University and Institut Fourier)

Factorial algebraic group actions and categorical quotients

(joint work with Devrim Celik and Juergen Hausen)

We begin with a survey of known results concerning categorical quotients.
Given an action of an affine algebraic group with only trivial characters on a factorial variety, we characterize existence of categorical quotient in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorial quotients for actions on, e.g. complete, varieties with finitely generated Cox ring via lifting to the universal torsor.

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