Seminar Algebra and Geometry: Émilie Dufresne (Universität Basel)

Finite separating sets and quasi-affine quotients

Nagata's famous counterexample to Hilbert's fourteenth problem shows that the ring of invariants of an algebraic group action on an affine algebraic variety is not always finitely generated. In some sense, however, invariant rings are not far from being affine. Indeed, invariant rings are always quasi-affine, and finite separating sets always exist. We give a new method for finding a quasi-affine variety on which the ring of regular functions is equal to a given invariant ring, and give a criterion to recognize separating algebras. We use the method and criterion to construct new examples.

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