Zero-Separating Invariants for Linear Algebraic Groups. Issue 4 (22nd December 2015) - Elmer, Jonathan; Kohls, Martin
Abstract: Abstract Let G be a linear algebraic group over an algebraically closed field 𝕜 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 ) = ∞ for any subgroup G of GL2 (𝕜) 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 0 is unipotent. Our results also lead to a more elementary proof that β sep ( G ) is finite if and only if G is finite.