#### Vol. 8, No. 8, 2014

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Proper triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are translations

Vol. 8 (2014), No. 8, 1959–1984
##### Abstract

We describe the structure of geometric quotients for proper locally triangulable ${\mathbb{G}}_{a}$-actions on locally trivial ${\mathbb{A}}^{3}$-bundles over a nœtherian normal base scheme $X$ defined over a field of characteristic $0$. In the case where $dimX=1$, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank $2$ over $X$. As a consequence, every proper triangulable ${\mathbb{G}}_{a}$-action on the affine four space ${\mathbb{A}}_{k}^{4}$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to ${\mathbb{A}}_{k}^{3}$.

##### Keywords
proper additive group actions, geometric quotients, principal homogeneous bundles, affine fibrations
##### Mathematical Subject Classification 2010
Primary: 14L30
Secondary: 14R10, 14R20, 14R25