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Non-linearizable algebraic $$k^*$$-actions on affine spaces. (English) Zbl 0933.14027
Let $$k^{[n]}$$ be a polynomial ring in $$n$$ variables over a field $$k$$. The linearity problem of torus actions asks if any faithful algebraic $$(k^*)^r$$-action of a torus on a $$k$$-affine space $$\mathbb{A}^n (=\text{Spec} k^{[n] })$$ is always linearizable, i.e. linear after a suitable change of the coordinate system of $$\mathbb{A}^n$$. The main purpose of the present paper is to give non-linearizable algebraic $$(k^*)^r$$-actions of torus on $$\mathbb{A}^n$$ in the following two cases:
(a) $$k$$ is an infinite field of characteristic $$p>0$$ for $$n>3$$ and $$r=1,\dots,n-2$$,
(b) $$k$$ is the real field $$\mathbb{R}$$ for $$n>4$$ and $$r=1,\dots,n-3$$.
As an application of the proof of these results we also give examples of nonsingular affine surfaces $$V$$ and $$W$$ such that $$V\times \mathbb{A}^1\cong_kW\times \mathbb{A}^1$$, but $$V\ncong_kW$$ for an arbitrary field $$k$$. Moreover if $$k$$ is algebraically closed, we show that some pairs of such $$V$$ and $$W$$ satisfy not only $$X\ncong_k Y$$ but also $$\operatorname{Aut}_k X\ncong \operatorname{Aut}_kY$$, where $$\operatorname{Aut}_kX$$ and $$\operatorname{Aut}_kY$$ denote the groups of all $$k$$-automorphisms of $$X$$ and $$Y$$, respectively.
Reviewer: T.Asanuma (Toyama)

##### MSC:
 14L30 Group actions on varieties or schemes (quotients) 20G05 Representation theory for linear algebraic groups 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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