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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References 
Similar Articles 
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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).
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Tommaso de Fernex, Birational transformations of prime order of the projective plane, Preprint 2001.
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I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque, Vol. 165 (1988). MR 90i:14009
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M. H. Gizatullin, Rational surfaces, Math. USSR Izv. 16 (1981), 103134. MR 81d:14020
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R. V. Gurjar, C. R. Pradeep, and D.Q. Zhang, On Gorenstein surfaces isomorphic to , Nagoya Math. J. to appear, math.AG/0112242.
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T. Hosoh, Automorphism groups of cubic surfaces, J. Algebra 192 (1997), 651677. MR 99d:14042
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T. Hosoh, Automorphism groups of quartic del Pezzo surfaces, J. Algebra 185 (1996), 374389. MR 97i:14026
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V. A. Iskovskih, Minimal models of rational surfaces over arbitrary fields, Math. USSR Izv. 14 (1981), 1739. MR 80m:14021
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S. Kantor, Theorie der endlichen Gruppen von eindeutigen Transformationen in der Ebene, Berlin: Mayer Muller, 1895.
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Y. Kawamata, A generalization of KodairaRamanujam's vanishing theorem, Math. Ann. 261 (1982), 4346. MR 84i:14022
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M. Koitabashi, Automorphism groups of generic rational surfaces, J. Algebra 116 (1988), 130 142. MR 89f:14045
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J. Kollar and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134 (1998). MR 2000b:14018
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Yu. I. Manin, Rational surfaces over perfect fields, II, Math. USSR Sb. 1 (1967), 141168. MR 37:1374
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Yu. I. Manin, Cubic forms : Algebra, geometry, arithmetic, 2nd ed, NorthHolland Math. Library, 4 (1986), NorthHolland Publ. Co., AmsterdamNew York. MR 87d:11037
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M. Miyanishi and K. Masuda, Open algebraic surfaces with finite group actions, Transform. Group, to appear.
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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.
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K. Oguiso, Automorphism groups in a family of K3 surfaces, math.AG/0104049.
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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.
 [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
