Proper triangular 𝔾a-actions on 𝔸4 are translations

Dubouloz, Adrien; Finston, David; Jaradat, Imad

doi:10.2140/ant.2014.8.1959

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

Adrien Dubouloz, David R. Finston and Imad Jaradat

Vol. 8 (2014), No. 8, 1959–1984

DOI: 10.2140/ant.2014.8.1959

Abstract

We describe the structure of geometric quotients for proper locally triangulable $G_{a}$ -actions on locally trivial $A^{3}$ -bundles over a nœtherian normal base scheme $X$ defined over a field of characteristic $0$ . In the case where $dim X = 1$ , 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 $G_{a}$ -action on the affine four space $A_{k}^{4}$ over a field of characteristic $0$ is a translation with geometric quotient isomorphic to $A_{k}^{3}$ .

Keywords

proper additive group actions, geometric quotients, principal homogeneous bundles, affine fibrations

Mathematical Subject Classification 2010

Primary: 14L30

Secondary: 14R10, 14R20, 14R25

Milestones

Received: 23 April 2014

Accepted: 10 September 2014

Published: 28 November 2014

Authors

Adrien Dubouloz
CNRS, Institut de Mathématiques de Bourgogne Université de Bourgogne 9 Avenue Alain Savary BP 47870 21078 Dijon France

David R. Finston
Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003 United States

Imad Jaradat
Department of Mathematical Sciences Jordan University of Science and Technology P.O.Box 3030 Irbid 22110 Jordan

Vol. 8, No. 8, 2014