On Hilbert's fourteenth problem for varieties of complexity one. (Q1319337)

Let \(G\) be a reductive group and \(B\) a Borel subgroup. The complexity \(c\) of a \(G\)-variety \(X\) is the codimension of a generic \(B\)-orbit. It is shown that the algebra of global functions \(k[X]\) is finitely generated whenever \(c\leq 1\) and \(X\) is unirational. Examples show, that none of the conditions can be relaxed. An application to the Popov-Pommerening conjecture, concerning radical subgroups, is given.