What is the correct combinatorial object to encode a linear representation? Many shadows of this problem have been studied:moment polytopes, Duistermaat-Heckman measures, Okounkov bodies. We suggest that already in very simple cases these miss a crucial feature. The ring theory, as opposed to just the linear algebra, of the group action on the coordinate ring, depends on some non-trivial lattice geometry and an associated filtration. Some striking similarities to, and key differences from, the theory of toric varieties ensue. Finite and non-finite generation phenomena emerge naturally. We discuss motivations from, and applications to, questions in the effective geometry of moduli of curves.
- Geometry and Analysis Seminar