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Online ISSN 1534-7486; Print ISSN 1056-3911
On automorphism groups of smooth hypersurfaces
Authors:
Song Yang, Xun Yu and Zigang Zhu
Journal:
J. Algebraic Geom. 34 (2025), 579-611
DOI:
https://doi.org/10.1090/jag/845
Published electronically:
January 29, 2025
Full-text PDF
Abstract | References | Additional Information
Abstract: We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few exceptions. For the exceptions, we give explicitly the defining equations and automorphism groups.
- Aldo Andreotti, Sopra le superficie algebriche che posseggono trasformazioni birazionali in sè, Univ. Roma Ist. Naz. Alta Mat. Rend. Mat. e Appl. (5) 9 (1950), 255–279 (Italian). MR 49601
- Jose Avila, Guillermo Ortiz, and Sergio Troncoso, Invariant smooth quartic surfaces by all finite primitive subgroups of $\textrm {PGL}_4(\Bbb {C})$, J. Pure Appl. Algebra 228 (2024), no. 4, Paper No. 107534, 24. MR 4649345, DOI 10.1016/j.jpaa.2023.107534
- E. Badr and F. Bars, The stratification by automorphism groups of smooth plane sextic curves, arXiv:2208.12749, 2022.
- H. Blichfeldt, Finite collineation groups, University of Chicago Press, Chicago, 1917.
- Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
- Richard Brauer, Über endliche lineare Gruppen von Primzahlgrad, Math. Ann. 169 (1967), 73–96 (German). MR 206088, DOI 10.1007/BF01399532
- W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 69818
- Fabrizio Catanese and Michael Schneider, Polynomial bounds for abelian groups of automorphisms, Compositio Math. 97 (1995), no. 1-2, 1–15. Special issue in honour of Frans Oort. MR 1355114
- H. C. Chang, On plane algebraic curves, Chinese J. Math. 6 (1978), no. 2, 185–189. MR 529972
- Ivan Cheltsov and Constantin Shramov, Finite collineation groups and birational rigidity, Selecta Math. (N.S.) 25 (2019), no. 5, Paper No. 71, 68. MR 4036497, DOI 10.1007/s00029-019-0516-5
- Michael J. Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007), no. 4, 411–423. MR 2334748, DOI 10.1515/JGT.2007.032
- Michael J. Collins, Bounds for finite primitive complex linear groups, J. Algebra 319 (2008), no. 2, 759–776. MR 2381807, DOI 10.1016/j.jalgebra.2005.11.042
- Michael J. Collins, Modular analogues of Jordan’s theorem for finite linear groups, J. Reine Angew. Math. 624 (2008), 143–171. MR 2456628, DOI 10.1515/CRELLE.2008.084
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
- Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
- I. Dolgachev, A brief introduction to automorphisms of algebraic varieties, Talca, Chile, 2019. https://dept.math.lsa.umich.edu/~idolga/chile1.pdf.
- I. Dolgachev and A. Duncan, Automorphisms of cubic surfaces in positive characteristic, Izv. Ross. Akad. Nauk Ser. Mat. 83 (2019), no. 3, 15–92; English transl., Izv. Math. 83 (2019), no. 3, 424–500. MR 3954305, DOI 10.4213/im8803
- Louis Esser, Automorphisms of weighted projective hypersurfaces, J. Pure Appl. Algebra 228 (2024), no. 6, Paper No. 107628, 21. MR 4697991, DOI 10.1016/j.jpaa.2024.107628
- L. Esser and J. Li, Hypersurfaces with large automorphism groups, arXiv:2405.11666v1, 2024.
- Walter Feit, The current situation in the theory of finite simple groups, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 55–93. MR 427449
- Walter Feit, On finite linear groups in dimension at most 10, Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) Academic Press, New York-London, 1976, pp. 397–407. MR 412294
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
- Víctor González-Aguilera and Alvaro Liendo, Automorphisms of prime order of smooth cubic $n$-folds, Arch. Math. (Basel) 97 (2011), no. 1, 25–37. MR 2820585, DOI 10.1007/s00013-011-0247-0
- Víctor González-Aguilera and Alvaro Liendo, On the order of an automorphism of a smooth hypersurface, Israel J. Math. 197 (2013), no. 1, 29–49. MR 3096604, DOI 10.1007/s11856-012-0177-y
- Víctor González-Aguilera, Alvaro Liendo, and Pedro Montero, On the liftability of the automorphism group of smooth hypersurfaces of the projective space, Israel J. Math. 255 (2023), no. 1, 283–310. MR 4619537, DOI 10.1007/s11856-022-2417-0
- Víctor González-Aguilera, Alvaro Liendo, Pedro Montero, and Roberto Villaflor Loyola, On a Torelli principle for automorphisms of Klein hypersurfaces, Trans. Amer. Math. Soc. 377 (2024), no. 8, 5483–5511. MR 4771229, DOI 10.1090/tran/9111
- Christopher D. Hacon, James McKernan, and Chenyang Xu, On the birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), no. 3, 1077–1111. MR 3034294, DOI 10.4007/annals.2013.177.3.6
- Takeshi Harui, Automorphism groups of smooth plane curves, Kodai Math. J. 42 (2019), no. 2, 308–331. MR 3981307, DOI 10.2996/kmj/1562032832
- Toshio Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), no. 2, 651–677. MR 1452681, DOI 10.1006/jabr.1996.6968
- Alan Howard and Andrew John Sommese, On the orders of the automorphism groups of certain projective manifolds, Manifolds and Lie groups (Notre Dame, Ind., 1980) Progr. Math., vol. 14, Birkhäuser, Boston, MA, 1981, pp. 145–158. MR 642855
- Daniel Huybrechts, The geometry of cubic hypersurfaces, Cambridge Studies in Advanced Mathematics, vol. 206, Cambridge University Press, Cambridge, 2023. MR 4589520, DOI 10.1017/9781009280020
- M. Camille Jordan, Mémoire sur les équations différentielles linéaires à intégrale algébrique, J. Reine Angew. Math. 84 (1878), 89–215. MR 1581645, DOI 10.1515/crelle-1878-18788408
- Radu Laza and Zhiwei Zheng, Automorphisms and periods of cubic fourfolds, Math. Z. 300 (2022), no. 2, 1455–1507. MR 4363785, DOI 10.1007/s00209-021-02810-x
- J. H. Lindsey II, Finite linear groups of degree six, Canadian J. Math. 23 (1971), 771–790. MR 289665, DOI 10.4153/CJM-1971-086-x
- Hideyuki Matsumura and Paul Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/64), 347–361. MR 168559, DOI 10.1215/kjm/1250524785
- G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and applications of finite groups, Dover Publications, Inc., New York, 1961. MR 123600
- Keiji Oguiso, A characterization of the Fermat quartic $K3$ surface by means of finite symmetries, Compos. Math. 141 (2005), no. 2, 404–424. MR 2134274, DOI 10.1112/S0010437X04001010
- Keiji Oguiso and Xun Yu, Automorphism groups of smooth quintic threefolds, Asian J. Math. 23 (2019), no. 2, 201–256. MR 3978250, DOI 10.4310/AJM.2019.v23.n2.a2
- Peter Orlik and Louis Solomon, Singularities. II. Automorphisms of forms, Math. Ann. 231 (1977/78), no. 3, 229–240. MR 476735, DOI 10.1007/BF01420243
- Fernanda Pambianco, Characterization of the Fermat curve as the most symmetric nonsingular algebraic plane curve, Math. Z. 277 (2014), no. 3-4, 975–993. MR 3229975, DOI 10.1007/s00209-014-1288-4
- Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623, DOI 10.1007/978-1-4612-4176-8
- B. Segre, The Non-singular Cubic Surfaces, Oxford University Press, Oxford, 1942. MR 8171
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- Tetsuji Shioda, Arithmetic and geometry of Fermat curves, Algebraic Geometry Seminar (Singapore, 1987) World Sci. Publishing, Singapore, 1988, pp. 95–102. MR 966448
- Endre Szabó, Bounding automorphism groups, Math. Ann. 304 (1996), no. 4, 801–811. MR 1380456, DOI 10.1007/BF01446320
- J. A. Todd, The invariants of a finite collineation group in five dimensions, Proc. Cambridge Philos. Soc. 46 (1950), 73–90. MR 34382, DOI 10.1017/s0305004100025494
- Gang Xiao, Bound of automorphisms of surfaces of general type. I, Ann. of Math. (2) 139 (1994), no. 1, 51–77. MR 1259364, DOI 10.2307/2946627
- Gang Xiao, Bound of automorphisms of surfaces of general type. II, J. Algebraic Geom. 4 (1995), no. 4, 701–793. MR 1339845
- Li Wei and Xun Yu, Automorphism groups of smooth cubic threefolds, J. Math. Soc. Japan 72 (2020), no. 4, 1327–1343. MR 4165935, DOI 10.2969/jmsj/83088308
- Song Yang, Xun Yu, and Zigang Zhu, Automorphism groups of cubic fivefolds and fourfolds, J. Lond. Math. Soc. (2) 110 (2024), no. 4, Paper No. e12997, 35. MR 4801899, DOI 10.1112/jlms.12997
- S. Yang, X. Yu, and Z. Zhu, On automorphism groups of smooth hypersurfaces, arXiv:2405.09505v1, 2024.
- Zhiwei Zheng, On abelian automorphism groups of hypersurfaces, Israel J. Math. 247 (2022), no. 1, 479–498. MR 4425345, DOI 10.1007/s11856-021-2275-1
Additional Information
Song Yang
Affiliation:
Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin 300072, People’s Republic of China
MR Author ID:
122624
Email:
syangmath@tju.edu.cn
Xun Yu
Affiliation:
Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin 300072, People’s Republic of China
MR Author ID:
1178446
Email:
xunyu@tju.edu.cn
Zigang Zhu
Affiliation:
Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin 300072, People’s Republic of China
MR Author ID:
1636249
ORCID:
0009-0000-6955-3277
Email:
zhzg0313@tju.edu.cn
Received by editor(s):
June 8, 2024
Received by editor(s) in revised form:
September 24, 2024, October 28, 2024, November 6, 2024, November 26, 2024, and December 4, 2024
Published electronically:
January 29, 2025
Additional Notes:
This work was partially supported by the National Natural Science Foundation of China (No. 12171351, No. 12071337, No. 11921001).
Article copyright:
© Copyright 2025
University Press, Inc.