## Additive actions on toric varieties

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- by Ivan Arzhantsev and Elena Romaskevich PDF
- Proc. Amer. Math. Soc.
**145**(2017), 1865-1879 Request permission

## 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.## References

- Ivan V. Arzhantsev,
*Flag varieties as equivariant compactifications of $\Bbb G^n_a$*, Proc. Amer. Math. Soc.**139**(2011), no. 3, 783–786. MR**2745631**, DOI 10.1090/S0002-9939-2010-10723-2 - Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface,
*Cox rings*, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015. MR**3307753** - I. V. Arzhantsev, M. G. Zaĭdenberg, and K. G. Kuyumzhiyan,
*Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity*, Mat. Sb.**203**(2012), no. 7, 3–30 (Russian, with Russian summary); English transl., Sb. Math.**203**(2012), no. 7-8, 923–949. MR**2986429**, DOI 10.1070/SM2012v203n07ABEH004248 - Ivan Arzhantsev, Alexander Perepechko, and Hendrik Süß,
*Infinite transitivity on universal torsors*, J. Lond. Math. Soc. (2)**89**(2014), no. 3, 762–778. MR**3217648**, DOI 10.1112/jlms/jdt081 - Ivan Arzhantsev and Andrey Popovskiy,
*Additive actions on projective hypersurfaces*, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., vol. 79, Springer, Cham, 2014, pp. 17–33. MR**3229343**, DOI 10.1007/978-3-319-05681-4_{2} - Ivan V. Arzhantsev and Elena V. Sharoyko,
*Hassett-Tschinkel correspondence: modality and projective hypersurfaces*, J. Algebra**348**(2011), 217–232. MR**2852238**, DOI 10.1016/j.jalgebra.2011.09.026 - Victor V. Batyrev and Yuri Tschinkel,
*Manin’s conjecture for toric varieties*, J. Algebraic Geom.**7**(1998), no. 1, 15–53. MR**1620682** - Antoine Chambert-Loir and Yuri Tschinkel,
*On the distribution of points of bounded height on equivariant compactifications of vector groups*, Invent. Math.**148**(2002), no. 2, 421–452. MR**1906155**, DOI 10.1007/s002220100200 - Antoine Chambert-Loir and Yuri Tschinkel,
*Integral points of bounded height on partial equivariant compactifications of vector groups*, Duke Math. J.**161**(2012), no. 15, 2799–2836. MR**2999313**, DOI 10.1215/00127094-1813638 - David A. Cox,
*The homogeneous coordinate ring of a toric variety*, J. Algebraic Geom.**4**(1995), no. 1, 17–50. MR**1299003** - David A. Cox, John B. Little, and Henry K. Schenck,
*Toric varieties*, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR**2810322**, DOI 10.1090/gsm/124 - Michel Demazure,
*Sous-groupes algébriques de rang maximum du groupe de Cremona*, Ann. Sci. École Norm. Sup. (4)**3**(1970), 507–588 (French). MR**284446** - U. Derenthal and D. Loughran,
*Singular del Pezzo surfaces that are equivariant compactifications*, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)**377**(2010), no. Issledovaniya po Teorii Chisel. 10, 26–43, 241 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.)**171**(2010), no. 6, 714–724. MR**2753646**, DOI 10.1007/s10958-010-0174-9 - Rostislav Devyatov,
*Unipotent commutative group actions on flag varieties and nilpotent multiplications*, Transform. Groups**20**(2015), no. 1, 21–64. MR**3317794**, DOI 10.1007/s00031-015-9306-0 - Evgeny Feigin,
*$\Bbb {G}_a^M$ degeneration of flag varieties*, Selecta Math. (N.S.)**18**(2012), no. 3, 513–537. MR**2960025**, DOI 10.1007/s00029-011-0084-9 - Gene Freudenburg,
*Algebraic theory of locally nilpotent derivations*, Encyclopaedia of Mathematical Sciences, vol. 136, Springer-Verlag, Berlin, 2006. Invariant Theory and Algebraic Transformation Groups, VII. MR**2259515** - Baohua Fu and Jun-Muk Hwang,
*Uniqueness of equivariant compactifications of $\Bbb C^n$ by a Fano manifold of Picard number 1*, Math. Res. Lett.**21**(2014), no. 1, 121–125. MR**3247043**, DOI 10.4310/MRL.2014.v21.n1.a9 - William Fulton,
*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037**, DOI 10.1515/9781400882526 - Brendan Hassett and Yuri Tschinkel,
*Geometry of equivariant compactifications of $\textbf {G}_a^n$*, Internat. Math. Res. Notices**22**(1999), 1211–1230. MR**1731473**, DOI 10.1155/S1073792899000665 - James E. Humphreys,
*Linear algebraic groups*, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR**0396773** - Alvaro Liendo,
*Affine $\Bbb T$-varieties of complexity one and locally nilpotent derivations*, Transform. Groups**15**(2010), no. 2, 389–425. MR**2657447**, DOI 10.1007/s00031-010-9089-2 - Tadao Oda,
*Convex bodies and algebraic geometry*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR**922894** - E. V. Sharoĭko,
*The Hassett-Tschinkel correspondence and automorphisms of a quadric*, Mat. Sb.**200**(2009), no. 11, 145–160 (Russian, with Russian summary); English transl., Sb. Math.**200**(2009), no. 11-12, 1715–1729. MR**2590000**, DOI 10.1070/SM2009v200n11ABEH004056

## Additional Information

**Ivan Arzhantsev**- Affiliation: Faculty of Computer Science, National Research University Higher School of Economics, Kochnovskiy Proezd 3, Moscow, 125319 Russia
- MR Author ID: 359575
- Email: arjantsev@hse.ru
**Elena Romaskevich**- Affiliation: Yandex, ulica L’va Tolstogo 16, Moscow, 119034 Russia
- MR Author ID: 1047535
- Email: lena.apq@gmail.com
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3611303