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Additive actions on toric varieties. (English) Zbl 1375.14155
The main object of the paper under review is an irreducible algebraic variety \(X\) of dimension \(n\) (defined over an algebraically closed field of characteristic zero) admitting an action of the commutative unipotent group \(\mathbb G^n_a\) (for short, an additive action). The study of such actions goes back to a paper by B. Hassett and Yu. Tschinkel [Internat. Math. Res. Notices 22, 1211–1230 (1999; Zbl. 0966.14033)] who established, in a particular case where \(X\) is the projective space, a correspondence between such actions and local \((n + 1)\)-dimensional commutative associative algebras with unit and classified such actions for small \(n\). Since then many other cases have been settled, and the authors present a brief survey along with a short description of arithmetic motivation. Their focus is on the case where \(X\) is an arbitrary toric variety. In this case, the first step is to consider actions normalized by the torus \(T\) acting on \(X\) (for short, normalized actions). It is known that normalized \(\mathbb G_a\)-actions are in one-to-one correspondence with several objects of different nature: combinatorial (so-called Demazure roots of the fan \(\Sigma\) defining \(X\)), group-theoretic (certain \(\mathbb G_a\)-subgroups of automorphisms of the Cox ring of \(X\)), or ring-theoretic (homogeneous locally nilpotent derivations of the Cox ring).
The main results of the paper are the following ones. First, Theorem 3.4 asserts that additive actions on a toric variety \(X\) normalized by the acting torus \(T\) are in bijection with so-called complete collections of Demazure roots of the fan \(\Sigma\). Second, they prove in Theorem 3.6 that such a normalized action is unique up to conjugation by an automorphism of \(X\). Third, they consider arbitrary additive actions, show that \(X\) admits a normalized action whenever it admits an arbitrary one, and provide a simple geometric criterion for such a situation. Furthermore, for the case where \(X\) is projective, they provide an elegant criterion for the existence of an additive action in terms of the corresponding polytope. They finish by considering explicit examples and open questions.

MSC:
14L30 Group actions on varieties or schemes (quotients)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13N15 Derivations and commutative rings
14J50 Automorphisms of surfaces and higher-dimensional varieties
14M17 Homogeneous spaces and generalizations
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References:
[1] Arzhantsev, Ivan V., Flag varieties as equivariant compactifications of \(\mathbb{G}^n_a\), Proc. Amer. Math. Soc., 139, 3, 783-786, (2011) · Zbl 1217.14032
[2] Arzhantsev, Ivan; Derenthal, Ulrich; Hausen, J\"urgen; Laface, Antonio, Cox rings, Cambridge Studies in Advanced Mathematics 144, viii+530 pp., (2015), Cambridge University Press, Cambridge · Zbl 1360.14001
[3] Arzhantsev, I. V.; Za\u \i denberg, M. G.; Kuyumzhiyan, K. G., Flag varieties, toric varieties, and suspensions: three examples of infinite transitivity, Mat. Sb.. Sb. Math., 203 203, 7-8, 923-949, (2012) · Zbl 1311.14059
[4] Arzhantsev, Ivan; Perepechko, Alexander; S\"u\ss , Hendrik, Infinite transitivity on universal torsors, J. Lond. Math. Soc. (2), 89, 3, 762-778, (2014) · Zbl 1342.14105
[5] Arzhantsev, Ivan; Popovskiy, Andrey, Additive actions on projective hypersurfaces. Automorphisms in birational and affine geometry, Springer Proc. Math. Stat. 79, 17-33, (2014), Springer, Cham · Zbl 1326.14112
[6] Arzhantsev, Ivan V.; Sharoyko, Elena V., Hassett-Tschinkel correspondence: modality and projective hypersurfaces, J. Algebra, 348, 217-232, (2011) · Zbl 1248.14053
[7] Batyrev, Victor V.; Tschinkel, Yuri, Manin’s conjecture for toric varieties, J. Algebraic Geom., 7, 1, 15-53, (1998) · Zbl 0946.14009
[8] Chambert-Loir, Antoine; Tschinkel, Yuri, On the distribution of points of bounded height on equivariant compactifications of vector groups, Invent. Math., 148, 2, 421-452, (2002) · Zbl 1067.11036
[9] Chambert-Loir, Antoine; Tschinkel, Yuri, Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J., 161, 15, 2799-2836, (2012) · Zbl 1348.11055
[10] Cox, David A., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 1, 17-50, (1995) · Zbl 0846.14032
[11] Cox, David A.; Little, John B.; Schenck, Henry K., Toric varieties, Graduate Studies in Mathematics 124, xxiv+841 pp., (2011), American Mathematical Society, Providence, RI · Zbl 1223.14001
[12] Demazure, Michel, Sous-groupes alg\'ebriques de rang maximum du groupe de Cremona, Ann. Sci. \'Ecole Norm. Sup. (4), 3, 507-588, (1970) · Zbl 0223.14009
[13] Derenthal, U.; Loughran, D., Singular del Pezzo surfaces that are equivariant compactifications, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N.Y.), 377 171, 6, 714-724, (2010) · Zbl 1296.14016
[14] Devyatov, Rostislav, Unipotent commutative group actions on flag varieties and nilpotent multiplications, Transform. Groups, 20, 1, 21-64, (2015) · Zbl 1393.14047
[15] Feigin, Evgeny, \(\mathbb{G}_a^M\) degeneration of flag varieties, Selecta Math. (N.S.), 18, 3, 513-537, (2012) · Zbl 1267.14064
[16] Freudenburg, Gene, Algebraic theory of locally nilpotent derivations, Encyclopaedia of Mathematical Sciences 136, xii+261 pp., (2006), Springer-Verlag, Berlin · Zbl 1121.13002
[17] Fu, Baohua; Hwang, Jun-Muk, Uniqueness of equivariant compactifications of \(\mathbb{C}^n\) by a Fano manifold of Picard number 1, Math. Res. Lett., 21, 1, 121-125, (2014) · Zbl 1327.32030
[18] Fulton, William, Introduction to toric varieties, Annals of Mathematics Studies 131, xii+157 pp., (1993), Princeton University Press, Princeton, NJ · Zbl 0813.14039
[19] Hassett, Brendan; Tschinkel, Yuri, Geometry of equivariant compactifications of \(\textbf{G}_a^n\), Internat. Math. Res. Notices, 22, 1211-1230, (1999) · Zbl 0966.14033
[20] Humphreys, James E., Linear algebraic groups, xiv+247 pp., (1975), Springer-Verlag, New York-Heidelberg · Zbl 0325.20039
[21] Liendo, Alvaro, Affine \(\mathbb{T}\)-varieties of complexity one and locally nilpotent derivations, Transform. Groups, 15, 2, 389-425, (2010) · Zbl 1209.14050
[22] Oda, Tadao, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 15, viii+212 pp., (1988), Springer-Verlag, Berlin · Zbl 0628.52002
[23] Sharo\u \i ko, E. V., The Hassett-Tschinkel correspondence and automorphisms of a quadric, Mat. Sb.. Sb. Math., 200 200, 11-12, 1715-1729, (2009) · Zbl 1205.13030
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