Abstract
We classify three-dimensional singular cubic hypersurfaces with an action of a finite group G, which are not G-rational and have no birational structure of G-Mori fiber space with the base of positive dimension. Also we prove the \(\mathfrak {A}_{5}\)-birational superrigidity of the Segre cubic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, D., Wang, J.: Equivariant resolution of singularities in characteristic 0. Math. Res. Lett. 4(2–3), 427–433 (1997)
Avilov, A.: Automorphisms of threefolds that can be represented as an intersection of two quadrics. Sb. Math. 207(3), 315–330 (2016)
Avilov, A.A.: Existence of standard models of conic bundles over algebraically non-closed fields. Sb. Math. 205(12), 1683–1695 (2014)
Bruce, J.W., Wall, C.T.C.: On the classification of cubic surfaces. J. London Math. Soc. 19(2), 245–256 (1979)
Cheltsov, I.A.: Birationally rigid Fano varieties. Russian Math. Surveys 60(5), 875–965 (2005)
Cheltsov, I., Przyjalkowski, V., Shramov, C.: Quartic double solids with icosahedral symmetry. Eur. J. Math. 2(1), 96–119 (2016)
Cheltsov, I., Shramov, C.: Five embeddings of one simple group. Trans. Amer. Math. Soc. 366(3), 1289–1331 (2014)
Cheltsov, I., Shramov, C.: Three embeddings of the Klein simple group into the Cremona group of rank three. Transform. Groups 17(2), 303–350 (2012)
Cheltsov, I., Shramov, C.: Cremona Groups and the Icosahedron. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016)
Cheltsov, I., Shramov, C.: Finite collineation groups and birational rigidity (2017). arXiv:1712.08258
Clemens, C.H., Griffiths, Ph.A.: The intermediate Jacobian of the cubic threefold. Ann. Math. 95(2), 281–356 (1972)
Corti, A.: Singularities of linear systems and 3-fold birational geometry. In: Corti, A., Reid, M. (eds.) Explicit Birational Geometry of 3-Folds. London Mathematical Society Lecture Note Series, vol. 281, pp. 259–312. Cambridge University Press, Cambridge (2000)
Dolgachev, I.V.: Classical Algebraic Geometry. Cambridge University Press, Cambridge (2012)
Dolgachev, I.V., Iskovskikh, V.A.: Finite subgroups of the plane Cremona group. In: Tschinkel, Yu., Zarhin, Yu. (eds.) Algebra, Arithmetic, and Geometry: in honor of Yu.I. Manin, vol. I. Progress in Mathematics, vol. 269, pp. 443–548. Birkhäuser, Boston (2009)
Finkelnberg, H., Werner, J.: Small resolutions of nodal cubic threefold. Nederl. Akad. Wetensch. Indag. Math. 51(2), 185–198 (1989)
Fujita, T.: On singular del Pezzo varieties. In: Sommese, A.J., Biancofiore, A., Livorni, E.L. (eds.) Algebraic Geometry. Lecture Notes in Mathematics, vol. 1417, pp. 117–128. Springer, Berlin (1990)
Fujita, T.: On the structure of polarized manifold with total deficiency one. I. J. Math. Soc. Japan 32(4), 709–725 (1980)
Fujita, T.: On the structure of polarized manifold with total deficiency one. II. J. Math. Soc. Japan 33(3), 415–434 (1981)
Fujita, T.: On the structure of polarized manifold with total deficiency one. III. J. Math. Soc. Japan 36(1), 75–89 (1984)
Fujita, T.: Classification Theories of Polarized Varieties. London Mathematical Society Lecture Note Series, vol. 155. Cambridge University Press, Cambridge (1990)
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998)
Kollár, J., Smith, K.E., Corti, A.: Rational and Nearly Rational Varieties. Cambridge Studies in Advanced Mathematics, vol. 92. Cambridge University Press, Cambridge (2004)
Matsuki, K.: Introduction to the Mori Program. Universitext. Springer, New York (2002)
Prokhorov, Yu.G.: Fields of invariants of finite linear groups. In: Bogomolov, F., Tschinkel, Yu. (eds.) Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 245–273. Birkhäuser, Boston (2010)
Prokhorov, Yu.: \(p\)-elementary subgroups of the Cremona group of rank 3. In: Faber, C., van der Geer, G., Looijenga, E. (eds.) Classification of Algebraic Varieties. EMS Series of Congress Reports, pp. 327–338. European Mathematical Society, Zürich (2011)
Prokhorov, Yu.: Simple finite subgroups of the Cremona group of rank 3. J. Algebraic Geom. 21(3), 563–600 (2012)
Prokhorov, Yu.: \(G\)-Fano threefolds. I. Adv. Geom. 13(3), 389–418 (2013)
Prokhorov, Yu.: 2-elementary subgroups of the space Cremona group. In: Cheltsov, I., et al. (eds.) Automorphisms in Birational and Affine Geometry, Springer Proceedings in Mathematics and Statistics, vol. 79, pp. 215–229. Springer, Cham (2014)
Prokhorov, Yu.G.: On \(G\)-Fano threefolds. Izv. Math. 79(4), 795–808 (2015)
Prokhorov, Yu.G.: Singular Fano manifolds of genus 12. Sb. Math. 207(7–8), 983–1009 (2016)
Reid, M.: Minimal models of canonical \(3\)-folds. In: Iitaka, S. (ed.) Algebraic Varieties and Analytic Varieties. Advanced Studies in Pure Mathematics, vol. 1, pp. 131–180. North-Holland, Amsterdam (1983)
Shepherd-Barron, N.I.: The Hasse principle for 9-nodal cubic threefolds (2012). arXiv:1210.5178
Teissier, B.: Cycles évanescents, sections planes et conditions de Whitney. In: Singularités à Cargèse. Astérisque, vols. 7, 8, 285–362. Société Mathématique de France, Paris (1973)
The GAP Group, GAP—Groups, Algorithms, and Programming. Version 4.8.2 (2016). http://www.gap-system.org
Acknowledgements
The author is a Young Russian Mathematics award winner and would like to thank its sponsors and jury. The author also would like to thank Ivan Cheltsov, Yuri Prokhorov, Constantin Shramov and Andrey Trepalin for useful discussions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by the Russian Science Foundation under Grant No. 18-11-00121.
Rights and permissions
About this article
Cite this article
Avilov, A. Automorphisms of singular three-dimensional cubic hypersurfaces. European Journal of Mathematics 4, 761–777 (2018). https://doi.org/10.1007/s40879-018-0253-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-018-0253-x