Zero-separating invariants for linear algebraic groups

Abstract

Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a $G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$ with the property that $f(v)=0$ for every positive degree homogeneous invariant $f$ in $k[V]^G$. We define numbers $δ(G,V)$ and $σ(G,V)$ associated with a given representation as follows: $δ(G,V)$ is the smallest number $d$ such that, for any point $v$ in $V^G$ outside the nullcone, there exists an invariant $f$ of degree at most $d$ such that $f(v)$ is not zero; $σ(G,V)$ is the same thing with $V^G$ replaced by $V$. If k has positive characteristic, we show that $δ(G,V)$ is infinite for all subgroups of $GL_2(k)$ containing a unipotent subgroup, and that $σ(G,V)$ is finite if and only if $G$ is finite. If $k$ has characteristic zero we show that $δ(G,V)=1$ for all linear algebraic groups and that if $σ(G,V)$ is finite then the connected component of $G$ is unipotent.