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Additive actions on projective hypersurfaces. (English) Zbl 1326.14112
Cheltsov, Ivan (ed.) et al., Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3, 2012. Cham: Springer (ISBN 978-3-319-05680-7/hbk; 978-3-319-05681-4/ebook). Springer Proceedings in Mathematics & Statistics 79, 17-33 (2014).
Summary: By an additive action on a hypersurface $$H$$ in $$\mathbb P^{n+1}$$ we mean an effective action of a commutative unipotent group on $$\mathbb P^{n+1}$$ which leaves $$H$$ invariant and acts on $$H$$ with an open orbit. B. Hassett and Y. Tschinkel [Int. Math. Res. Not. 1999, No. 22, 1211–1230 (1999; Zbl 0966.14033)] have shown that actions of commutative unipotent groups on projective spaces can be described in terms of local algebras with some additional data. We prove that additive actions on projective hypersurfaces correspond to invariant multilinear symmetric forms on local algebras. It allows us to obtain explicit classification results for non-degenerate quadrics and quadrics of corank one.
For the entire collection see [Zbl 1291.14005].

MSC:
 14L30 Group actions on varieties or schemes (quotients) 13H99 Local rings and semilocal rings 15A69 Multilinear algebra, tensor calculus 14J50 Automorphisms of surfaces and higher-dimensional varieties
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References:
 [1] I. Arzhantsev, Flag varieties as equivariant compactifications of · Zbl 1217.14032 · doi:10.1090/S0002-9939-2010-10723-2 [2] I. Arzhantsev, E. Sharoyko, Hassett-Tschinkel correspondence: Modality and projective hypersurfaces. J. Algebra 348(1), 217-232 (2011) · Zbl 1248.14053 · doi:10.1016/j.jalgebra.2011.09.026 [3] I. Bazhov, Additive structures on cubic hypersurfaces, 8 pp. [arXiv:1307.6085] · Zbl 1359.14004 [4] A. Chambert-Loir, Y. Tschinkel, On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148(2), 421-452 (2002) · Zbl 1067.11036 · doi:10.1007/s002220100200 [5] U. Derenthal, D. Loughran, Singular del Pezzo surfaces that are equivariant compactifications. J. Math. Sci. 171(6), 714-724 (2010) · Zbl 1296.14016 · doi:10.1007/s10958-010-0174-9 [6] U. Derenthal, D. Loughran, Equivariant compactifications of two-dimensional algebraic groups, to appear in: Proceedings of the Edinburgh Mathematical Society 19 pp. [arXiv:1212.3518] · Zbl 1368.14059 [7] R. Devyatov, Unipotent commutative group actions on flag varieties and nilpotent multiplications, 25 pp. [arXiv:1309.3480] · Zbl 1393.14047 [8] E. Feigin, [9] B. Fu, J.-M. Hwang, Uniqueness of equivariant compactifications of [10] F. Gantmacher, $$The Theory of Matrices$$, Transl. from Russian by K.A. Hirsch (American Mathematical Society, Providence, 2000) · Zbl 0927.15002 [11] B. Hassett, Y. Tschinkel, Geometry of equivariant compactifications of · Zbl 0966.14033 · doi:10.1155/S1073792899000665 [12] F. Knop, H. Kraft, D. Luna, T. Vust, Local properties of algebraic group actions, in $$Algebraische Transformationsgruppen und Invariantentheorie$$. DMV Seminar, vol. 13 (Birkhäuser, Basel, 1989), pp. 63-75 · Zbl 0722.14032 [13] H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. 3(3), 347-361 (1963) · Zbl 0141.37401 [14] V. Popov, E. Vinberg, Invariant theory, in $$Algebraic Geometry IV$$, ed. by A.N. Parshin, I.R. Shafarevich (Springer, Berlin, 1994) · Zbl 0735.14010 [15] E. Sharoiko, Hassett-Tschinkel correspondence and automorphisms of a quadric. Sb. Math. 200(11), 1715-1729 (2009) · Zbl 1205.13030 · doi:10.1070/SM2009v200n11ABEH004056 [16] D. Timashev, in $$Homogeneous Spaces and Equivariant Embeddings$$. Encyclopaedia of Mathematical Sciences, vol. 138 (Springer, Berlin, 2011) · Zbl 1237.14057
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