Observability in invariant theory

Abstract:

We consider actions $G\times X\to X$ of the affine, algebraic group $G$ on the affine, algebraic variety $X$. We say that $G\times X\to X$ is observable in codimension one if for any height-one, $G$-invariant, prime ideal $\mathfrak {p}\subset k[X]$, $\mathfrak {p}^G\neq (0)$. Many familiar actions are observable in codimension one. We characterize such actions geometrically and indicate how they fit into the general framework of invariant theory. We look at what happens if we impose further restrictions, such as $G$ being reductive or $X$ being factorial. We indicate how Grosshans subgroups are involved.