Abstract
We study induced additive actions on projective hypersurfaces, i.e. effective regular actions of the algebraic group \({\mathbb {G}}_a^m\) with an open orbit that can be extended to a regular action on the ambient projective space. We prove that if a projective hypersurface admits an induced additive action, then it is unique if and only if the hypersurface is non-degenerate. We also show that for any \(n\ge 2\), there exists a non-degenerate hypersurface in \(\mathbb {P}^n\) of each degree d from 2 to n.
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Acknowledgements
The author is grateful to Ivan Arzhantsev and Yulia Zaitseva for usefuls discussions.
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This work was supported by the Russian Science Foundation Grant 23-21-00472
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Beldiev, I. Gorenstein Algebras and Uniqueness of Additive Actions. Results Math 78, 192 (2023). https://doi.org/10.1007/s00025-023-01972-w
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DOI: https://doi.org/10.1007/s00025-023-01972-w