Zero-separating invariants for linear algebraic groups

Article, Preprint English OPEN
Elmer, Jonathan ; Kohls, Martin (2014)
  • Publisher: Cambridge University Press
  • Related identifiers: doi: 10.1017/S0013091515000322
  • Subject: 13A50 | Mathematics - Commutative Algebra | Mathematics - Algebraic Geometry

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.
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