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Additive actions on toric varieties


Authors: Ivan Arzhantsev and Elena Romaskevich
Journal: Proc. Amer. Math. Soc. 145 (2017), 1865-1879
MSC (2010): Primary 14L30, 14M25; Secondary 13N15, 14J50, 14M17
DOI: https://doi.org/10.1090/proc/13349
Published electronically: October 26, 2016
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Abstract: By an additive action on an algebraic variety $ X$ of dimension $ n$ we mean a regular action $ \mathbb{G}_a^n\times X\to X$ with an open orbit of the commutative unipotent group $ \mathbb{G}_a^n$. We prove that if a complete toric variety $ X$ admits an additive action, then it admits an additive action normalized by the acting torus. Normalized additive actions on a toric variety $ X$ are in bijection with complete collections of Demazure roots of the fan $ \Sigma _X$. Moreover, any two normalized additive actions on $ X$ are isomorphic.


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Additional Information

Ivan Arzhantsev
Affiliation: Faculty of Computer Science, National Research University Higher School of Economics, Kochnovskiy Proezd 3, Moscow, 125319 Russia
Email: arjantsev@hse.ru

Elena Romaskevich
Affiliation: Yandex, ulica L’va Tolstogo 16, Moscow, 119034 Russia
Email: lena.apq@gmail.com

DOI: https://doi.org/10.1090/proc/13349
Keywords: Toric variety, automorphism, unipotent group, locally nilpotent derivation, Cox ring, Demazure root
Received by editor(s): October 30, 2015
Received by editor(s) in revised form: June 26, 2016
Published electronically: October 26, 2016
Additional Notes: The research of the first author was supported by the grant RSF-DFG 16-41-01013.
Communicated by: Lev Borisov
Article copyright: © Copyright 2016 American Mathematical Society