Examples and Counterexamples for Existence of Categorical Quotients

We give examples for existence and non-existence of categorical quotients for algebraic group actions in the categories of algebraic varieties and prevarieties. All our examples are subtorus actions on toric varieties.Comment: 15 pages, 3 figures, LaTeX2e + pste

Orbit spaces of small tori

Consider an algebraic torus of small dimension acting on an open subset of a complex vector space, or more generally on a quasiaffine variety such that a separated orbit space exists. We discuss under which conditions this orbit space is quasiprojective. One of our counterexamples provides a toric variety with enough effective invariant Cartier divisors that is not embeddable into a smooth toric variety.Comment: 11 pages, 1 figur