Zero-separating invariants for linear algebraic groups

Elmer, Jonathan and Kohls, Martin (2016) Zero-separating invariants for linear algebraic groups. Proceedings of the Edinburgh Mathematical Society, 59 (4). pp. 911-924. ISSN 0013-0915

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Abstract

Let G be linear algebraic group over an algebraically closed field k acting rationally on a G-module V , and N(G,V) its nullcone. Let δ(G, V ) and σ(G, V ) denote the minimal number d, such that for any v ∈ V^G \ N(G,V) and v ∈ V \ N(G,V) 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 GL 2 (k) which 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.

Item Type: Article
Additional Information: Published online: 22 December 2015
Research Areas: A. > School of Science and Technology > Design Engineering and Mathematics
Item ID: 20904
Notes on copyright: This article has been published in a revised form in Proceedings of the Edinburgh Mathematical Society http://dx.doi.org/10.1017/S0013091515000322. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © Edinburgh Mathematical Society 2016
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Depositing User: Jonathan Elmer
Date Deposited: 04 Nov 2016 14:13
Last Modified: 04 Apr 2019 06:37
URI: https://eprints.mdx.ac.uk/id/eprint/20904

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