Abstract
Let G be a linear algebraic group over an algebraically closed field acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and, respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = â for any subgroup G of GL2() that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G is unipotent. Our results also lead to a more elementary proof that β sep(G) is finite if and only if G is finite.
| Original language | English |
|---|---|
| Pages (from-to) | 911-924 |
| Number of pages | 14 |
| Journal | Proceedings of the Edinburgh Mathematical Society |
| Volume | 59 |
| Issue number | 4 |
| Early online date | 22 Dec 2015 |
| DOIs | |
| Publication status | Published - 1 Nov 2016 |
Bibliographical note
This paper was prepared during visits of M.K. to the Universityof Aberdeen and of J.E. to T. U. M¨unchen. The first of these visits was supported by the Edinburgh Mathematical Society’s Research Support Fund. We want to thank Gregor Kemper and the Edinburgh Mathematical Society for making these visits possible. We also thank the anonymous referee for useful suggestions.
Keywords
- geometrically reductive
- global degree bounds
- invariant theory
- linear algebraic groups
- prime characteristic