Zero-separating invariants for linear algebraic groups

TL;DR

This paper investigates invariants in linear algebraic group actions, defining numerical measures of separating points from the nullcone, and characterizes their finiteness properties across different group types and field characteristics.

Contribution

It introduces and analyzes the invariants δ(G,V) and σ(G,V), providing new bounds and characterizations for their finiteness depending on the group's structure and the field's characteristic.

Findings

δ(G,V) is infinite for certain subgroups in positive characteristic

σ(G,V) is finite if and only if G is finite in positive characteristic

δ(G,V)=1 for all groups in characteristic zero

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 $\delta(G,V)$ and $\sigma(G,V)$ associated with a given representation as follows: $\delta(G,V)$ 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; $\sigma(G,V)$ is the same thing with $V^G$ replaced by $V$. If k has positive characteristic, we show that $\delta(G,V)$ is infinite for all subgroups of $GL_2(k)$ containing a unipotent subgroup, and that $\sigma(G,V)$ is finite if and only if $G$ is finite. If $k$ has characteristic zero we show that $\delta(G,V)=1$ for all linear algebraic groups and that if $\sigma(G,V)$ is finite then the connected component of $G$ is unipotent.