Automorphisms of finite order on Gorenstein del Pezzo surfaces

Author:
D.-Q. Zhang

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4831-4845

MSC (2000):
Primary 14J50; Secondary 14J26

DOI:
https://doi.org/10.1090/S0002-9947-02-03069-6

Published electronically:
August 1, 2002

MathSciNet review:
1926853

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall determine all actions of groups of prime order with on Gorenstein del Pezzo (singular) surfaces of Picard number 1. We show that every order- element in ( , being the minimal resolution of ) is lifted from a projective transformation of . We also determine when is finite in terms of , and the number of singular members in . In particular, we show that either for some , or for every prime , there is at least one element of order in (hence is infinite).

**[dF]**Tommaso de Fernex, Birational transformations of prime order of the projective plane, Preprint 2001.**[D]**Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.),*Séminaire sur les Singularités des Surfaces*, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980 (French). Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977. MR**579026****[DO]**Igor Dolgachev and David Ortland,*Point sets in projective spaces and theta functions*, Astérisque**165**(1988), 210 pp. (1989) (English, with French summary). MR**1007155****[G]**M. H. Gizatullin,*Rational 𝐺-surfaces*, Izv. Akad. Nauk SSSR Ser. Mat.**44**(1980), no. 1, 110–144, 239 (Russian). MR**563788****[GPZ]**R. V. Gurjar, C. R. Pradeep, and D.-Q. Zhang, On Gorenstein surfaces isomorphic to , Nagoya Math. J. to appear, math.AG/0112242.**[H1]**Toshio Hosoh,*Automorphism groups of cubic surfaces*, J. Algebra**192**(1997), no. 2, 651–677. MR**1452681**, https://doi.org/10.1006/jabr.1996.6968**[H2]**Toshio Hosoh,*Automorphism groups of quartic del Pezzo surfaces*, J. Algebra**185**(1996), no. 2, 374–389. MR**1417377**, https://doi.org/10.1006/jabr.1996.0331**[I]**V. A. Iskovskih,*Minimal models of rational surfaces over arbitrary fields*, Izv. Akad. Nauk SSSR Ser. Mat.**43**(1979), no. 1, 19–43, 237 (Russian). MR**525940****[K]**S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin: Mayer Muller, 1895.**[Ka]**Yujiro Kawamata,*A generalization of Kodaira-Ramanujam’s vanishing theorem*, Math. Ann.**261**(1982), no. 1, 43–46. MR**675204**, https://doi.org/10.1007/BF01456407**[Ko]**Masanori Koitabashi,*Automorphism groups of generic rational surfaces*, J. Algebra**116**(1988), no. 1, 130–142. MR**944150**, https://doi.org/10.1016/0021-8693(88)90196-2**[KM]**János Kollár and Shigefumi Mori,*Birational geometry of algebraic varieties*, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR**1658959****[M1]**Ju. I. Manin,*Rational surfaces over perfect fields. II*, Mat. Sb. (N.S.)**72 (114)**(1967), 161–192 (Russian). MR**0225781****[M2]**Yu. I. Manin,*Cubic forms*, 2nd ed., North-Holland Mathematical Library, vol. 4, North-Holland Publishing Co., Amsterdam, 1986. Algebra, geometry, arithmetic; Translated from the Russian by M. Hazewinkel. MR**833513****[MM]**M. Miyanishi and K. Masuda, Open algebraic surfaces with finite group actions, Transform. Group, to appear.**[MP]**Rick Miranda and Ulf Persson,*On extremal rational elliptic surfaces*, Math. Z.**193**(1986), no. 4, 537–558. MR**867347**, https://doi.org/10.1007/BF01160474**[MZ1]**M. Miyanishi and D.-Q. Zhang,*Gorenstein log del Pezzo surfaces of rank one*, J. Algebra**118**(1988), no. 1, 63–84. MR**961326**, https://doi.org/10.1016/0021-8693(88)90048-8**[MZ2]**M. Miyanishi and D.-Q. Zhang,*Gorenstein log del Pezzo surfaces. II*, J. Algebra**156**(1993), no. 1, 183–193. MR**1213791**, https://doi.org/10.1006/jabr.1993.1069**[MZ3]**M. Miyanishi and D. -Q. Zhang, Equivariant classification of Gorenstein open log del Pezzo surfaces with finite group actions, Preprint 2001.**[O]**K. Oguiso, Automorphism groups in a family of K3 surfaces, math.AG/0104049.**[OS]**Keiji Oguiso and Tetsuji Shioda,*The Mordell-Weil lattice of a rational elliptic surface*, Comment. Math. Univ. St. Paul.**40**(1991), no. 1, 83–99. MR**1104782****[S]**B. Segre, The non-singular cubic surfaces, Oxford University Press, Oxford, 1942. MR**4:254b****[V]**Eckart Viehweg,*Vanishing theorems*, J. Reine Angew. Math.**335**(1982), 1–8. MR**667459**, https://doi.org/10.1515/crll.1982.335.1**[Y1]**Q. Ye, On Gorenstein log del Pezzo surfaces, Japanese J. Math. to appear, math.AG / 0109223.**[Y2]**Q. Ye, On algebraic surfaces with non-positive Kodaira dimension, Ph.D. thesis, National Univ. of Singapore, 2001.**[ZD]**D.-Q. Zhang,*Automorphisms of finite order on rational surfaces*, J. Algebra**238**(2001), no. 2, 560–589. With an appendix by I. Dolgachev. MR**1823774**, https://doi.org/10.1006/jabr.2000.8673**[Z2]**D.-Q. Zhang, Automorphisms of finite order on extremal rational elliptic surfaces and Gorenstein del Pezzo surfaces of degree one, Preprint 2001.

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Additional Information

**D.-Q. Zhang**

Affiliation:
Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore

Email:
matzdq@math.nus.edu.sg

DOI:
https://doi.org/10.1090/S0002-9947-02-03069-6

Received by editor(s):
March 10, 2002

Published electronically:
August 1, 2002

Article copyright:
© Copyright 2002
American Mathematical Society