Abstract
In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of \({\mathbb{P}^{n+1}}\), for a fixed n ≥ 2. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p < 2n+1 and that any such hypersurface admitting an automorphism of order p > 2n is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli space of principally polarized abelian varieties.
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References
Adler A.: On the automorphism group of a certain cubic threefold. Amer. J. Math. 100, 1275–1280 (1978)
Bennama H., Bertin J.: Remarque sur les variétés abéliennes avec un automorphisme d’ordre premier. Manuscripta Math. 94, 409–425 (1997)
P. Deligne, Cohomologie des intersections complètes, Groupes de Monodromie en Géométrie Algebrique. SGA 7 II, Exposé XI, Springer Lectures Notes 340, 1973.
I. V. Dolgachev and A. Verra Topics in classical algebraic geometry, preprint, 2004.
González-Aguilera V., Muñoz-Porras J.M., Zamora A.G.R.: On the 0-dimensional irreducible components of the singular locus of \({\mathcal{A}_g}\) . Arch. Math. 84, 298–303 (2005)
Klein F.: Über die Ausflösung gewisser Gleichungen vom siebenten und achten Grade. Math. Ann. 15, 251–282 (1879)
Kontogeorgis A.: Automorphisms of Fermat-like varieties. Manuscripta math. 107, 187–205 (2002)
A. Liendo, Sobre hipersuperficies cúbicas de \({\mathbb{P}^n}\), Master thesis at the Universidad Técnica Federico Santa María, 2005.
Matsumura H., Monsky P.: On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. 3, 347–361 (1964)
Roulleau X.: The Fano surface of the Klein cubic threefold, J. Math. Kyoto Univ. 49, 113–129 (2009)
Szabo E.: Bounding automorphism groups. Math. Ann. 304, 801–811 (1996)
B. Segre, The non-singular cubic surfaces, Oxford at the Clarendon Press (1942).
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Víctor González-Aguilera was partially supported by the Fondecyt project 1080030 and the DGIP of the UTFSM.
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González-Aguilera, V., Liendo, A. Automorphisms of prime order of smooth cubic n-folds. Arch. Math. 97, 25–37 (2011). https://doi.org/10.1007/s00013-011-0247-0
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DOI: https://doi.org/10.1007/s00013-011-0247-0