Abstract
We give a direct and self-contained proof of the fact that additive group actions on affine four-space generated by certain types of triangular derivations are translations whenever they are proper. The argument, which is based on explicit techniques, provides an illustration of the difficulties encountered and an introduction to the more abstract methods which were used recently by the authors to solve the general triangular case.
Subject Classification: 14R20; 14L30
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Notes
- 1.
The factor (m + 1) in \(\partial (x_{3})\) is chosen to simplify calculations in the next sections.
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Acknowledgements
The authors wish to thank the anonymous referees for their very careful reading and valuable comments which helped to correct some inaccuracies in a previous version and contributed to improve the quality of the presentation.
Research supported in part by National Science Foundation Grant 0936691 and ANR Grant “BirPol” ANR-11-JS01-004-01.
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Dubouloz, A., Finston, D.R., Jaradat, I. (2014). Equivariant Triviality of Quasi-Monomial Triangular \(\mathbb{G}_{a}\)-Actions on \(\mathbb{A}^{4}\) . In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_16
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