Automorphisms of finite order on Gorenstein del Pezzo surfaces
Author:
D.Q. Zhang
Journal:
Trans. Amer. Math. Soc. 354 (2002), 48314845
MSC (2000):
Primary 14J50; Secondary 14J26
Published electronically:
August 1, 2002
MathSciNet review:
1926853
Fulltext PDF Free Access
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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
(82d:14021)
 [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
(90i:14009)
 [G]
M.
H. Gizatullin, Rational 𝐺surfaces, Izv. Akad. Nauk
SSSR Ser. Mat. 44 (1980), no. 1, 110–144, 239
(Russian). MR
563788 (81d:14020)
 [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
(99d:14042), http://dx.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
(97i:14026), http://dx.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
(80m:14021)
 [K]
S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin: Mayer Muller, 1895.
 [Ka]
Yujiro
Kawamata, A generalization of KodairaRamanujam’s vanishing
theorem, Math. Ann. 261 (1982), no. 1,
43–46. MR
675204 (84i:14022), http://dx.doi.org/10.1007/BF01456407
 [Ko]
Masanori
Koitabashi, Automorphism groups of generic rational surfaces,
J. Algebra 116 (1988), no. 1, 130–142. MR 944150
(89f:14045), http://dx.doi.org/10.1016/00218693(88)901962
 [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
(2000b:14018)
 [M1]
Ju.
I. Manin, Rational surfaces over perfect fields. II, Mat. Sb.
(N.S.) 72 (114) (1967), 161–192 (Russian). MR 0225781
(37 #1374)
 [M2]
Yu.
I. Manin, Cubic forms, 2nd ed., NorthHolland Mathematical
Library, vol. 4, NorthHolland Publishing Co., Amsterdam, 1986.
Algebra, geometry, arithmetic; Translated from the Russian by M.
Hazewinkel. MR
833513 (87d:11037)
 [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
(88a:14044), http://dx.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
(89i:14033), http://dx.doi.org/10.1016/00218693(88)900488
 [MZ2]
M.
Miyanishi and D.Q.
Zhang, Gorenstein log del Pezzo surfaces. II, J. Algebra
156 (1993), no. 1, 183–193. MR 1213791
(94m:14045), http://dx.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 MordellWeil lattice of a rational elliptic
surface, Comment. Math. Univ. St. Paul. 40 (1991),
no. 1, 83–99. MR 1104782
(92g:14036)
 [S]
B.
Segre, The Nonsingular Cubic Surfaces, Oxford University
Press, Oxford, 1942. MR 0008171
(4,254b)
 [V]
Eckart
Viehweg, Vanishing theorems, J. Reine Angew. Math.
335 (1982), 1–8. MR 667459
(83m:14011), http://dx.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 nonpositive 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
(2002c:14064), http://dx.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.
 [dF]
 Tommaso de Fernex, Birational transformations of prime order of the projective plane, Preprint 2001.
 [D]
 M. Demazure, Lecture Notes in Mathematics, 777 (1980), Springer. MR 82d:14021
 [DO]
 I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque, Vol. 165 (1988). MR 90i:14009
 [G]
 M. H. Gizatullin, Rational surfaces, Math. USSR Izv. 16 (1981), 103134. MR 81d:14020
 [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]
 T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651677. MR 99d:14042
 [H2]
 T. Hosoh, Automorphism groups of quartic del Pezzo surfaces, J. Algebra 185 (1996), 374389. MR 97i:14026
 [I]
 V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Math. USSR Izv. 14 (1981), 1739. MR 80m:14021
 [K]
 S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin: Mayer Muller, 1895.
 [Ka]
 Y. Kawamata, A generalization of KodairaRamanujam's vanishing theorem, Math. Ann. 261 (1982), 4346. MR 84i:14022
 [Ko]
 M. Koitabashi, Automorphism groups of generic rational surfaces, J. Algebra 116 (1988), 130 142. MR 89f:14045
 [KM]
 J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134 (1998). MR 2000b:14018
 [M1]
 Yu. I. Manin, Rational surfaces over perfect fields, II, Math. USSR Sb. 1 (1967), 141168. MR 37:1374
 [M2]
 Yu. I. Manin, Cubic forms : Algebra, geometry, arithmetic, 2nd ed, NorthHolland Math. Library, 4 (1986), NorthHolland Publ. Co., AmsterdamNew York. MR 87d:11037
 [MM]
 M. Miyanishi and K. Masuda, Open algebraic surfaces with finite group actions, Transform. Group, to appear.
 [MP]
 R. Miranda and U. Persson, On extremal rational elliptic surfaces, Math. Z. 193 (1986), 537558. MR 88a:14044
 [MZ1]
 M. Miyanishi and D.Q. Zhang, Gorenstein log del Pezzo surfaces of rank one, I; J. Algebra 118 (1988), 6384; 156 (1993), 183193. MR 89i:14033
 [MZ2]
 M. Miyanishi and D.Q. Zhang, Gorenstein log del Pezzo surfaces of rank one, II; J. Algebra 118 (1988), 6384; 156 (1993), 183193. MR 94m:14045
 [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]
 K. Oguiso and T. Shioda, The MordellWeil lattice of a rational elliptic surface, Comment. Math. Univ. St. Paul. 40 (1991), 8399. MR 92g:14036
 [S]
 B. Segre, The nonsingular cubic surfaces, Oxford University Press, Oxford, 1942. MR 4:254b
 [V]
 E. Viehweg, Vanishing theorems, J. Reine Angew. Math. 335 (1982), 18. MR 83m:14011
 [Y1]
 Q. Ye, On Gorenstein log del Pezzo surfaces, Japanese J. Math. to appear, math.AG / 0109223.
 [Y2]
 Q. Ye, On algebraic surfaces with nonpositive Kodaira dimension, Ph.D. thesis, National Univ. of Singapore, 2001.
 [ZD]
 D.Q. Zhang, Automorphisms of finite order on rational surfaces, with an appendix by I. Dolgachev, J. Algebra 238 (2001), 560589. MR 2002c:14064
 [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:
http://dx.doi.org/10.1090/S0002994702030696
PII:
S 00029947(02)030696
Received by editor(s):
March 10, 2002
Published electronically:
August 1, 2002
Article copyright:
© Copyright 2002
American Mathematical Society
