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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The moduli space of Enriques surfaces and Borcherds products


Author: Shigeyuki Kondo
Journal: J. Algebraic Geom. 11 (2002), 601-627
DOI: https://doi.org/10.1090/S1056-3911-02-00301-6
Published electronically: March 18, 2002
MathSciNet review: 1910262
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Abstract | References | Additional Information

Abstract: We shall give an $O^{+}(10, \mathbf{F}_{2})$-equivariant birational holomorphic map from the moduli space of Enriques surfaces with level 2 structure to $\mathbf{P}^{185}$ by using Borcherds' theory of automorphic forms on a bounded symmetric domain of type IV. Its image satisfies $2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 7 \cdot 17 \cdot 31$ quartic relations.


References [Enhancements On Off] (What's this?)

  • [A] D. Allcock, The period lattice for Enriques surfaces, Math. Ann., 317(2000), 483-488.
  • [AF] D. Allcock, E. Freitag, Cubic surfaces and Borcherds product, math.AG/0002066.
  • [BB] W.L. Baily, Jr., A. Borel, A compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84(1966), 422-528.
  • [Ba] H.F. Baker, Principles of geometry II, Cambridge University Press 1922.
  • [BP] W. Barth, C. Peters, Automorphisms of Enriques surfaces, Invent. Math., 73 (1983), 383-411.
  • [B1] R. Borcherds, The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996), 699-710.
  • [B2] R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math., 132 (1998), 491-562.
  • [B3] R. Borcherds, An automorphic form related to cubic surfaces, unpublished manuscript, math.AG/0002079.
  • [C] J.H. Conway et al., Atlas of Finite Groups, Oxford Univ., Oxford 1985.
  • [Di] J. Dieudonné, La géométrie des groupes classiques (2nd ed.), Springer 1963.
  • [Do] I. Dolgachev, Enriques surfaces : what is left ?, Problems in the theory of surfaces and their classification (Cortona, 1988), 81-97, Sympos. Math., XXXII, Academic Press, London 1991.
  • [F] E. Freitag, Some modular forms related to cubic surfaces, preprint, August 1999.
  • [FH] E. Freitag, C.F. Hermann, Some modular varieties of low dimension, Adv. Math., 152 (2000), 203-287.
  • [Hi] D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313-373.
  • [Ho] E. Horikawa, On the periods of Enriques surfaces I, II, Math. Ann., 234 (1978), 78-108, ibid 235 (1978), 217-246.
  • [I] J. Igusa, On the graded ring of theta-constants, Amer. J. Math., 86 (1964), 219-246.
  • [K1] S. Kondo, Enriques surfaces with finite automorphism groups, Japan. J. Math., 12 (1986), 191-282.
  • [K2] S. Kondo, The rationality of the moduli space of Enriques surfaces, Compositio Math., 91 (1994), 159-173.
  • [MN] S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on cohomology groups, Invent. Math., 77 (1984), 383-397.
  • [Na] Y. Namikawa, Periods of Enriques surfaces, Math. Ann., 270 (1985), 201-222.
  • [N1] V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv., 14 (1980), 103-167.
  • [N2] V.V. Nikulin, On a description of the automorphism groups of Enriques surfaces, Soviet Math. Dokl., 30 (1984), 282-285.
  • [S] H. Sterk, Compactifications of the period space of Enriques surfaces I, II, Math. Z., 207 (1991), 1-36, ibid 220 (1995), 427-444.


Additional Information

Shigeyuki Kondo
Affiliation: Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602 Japan
Email: kondo@math.nagoya-u.ac.jp

DOI: https://doi.org/10.1090/S1056-3911-02-00301-6
Received by editor(s): May 18, 2000
Received by editor(s) in revised form: October 18, 2000
Published electronically: March 18, 2002
Additional Notes: Partially supported by Grants-in-Aid for Scientific Research (B)(2):10440005 and Houga: 11874004, Ministry of Education, Science and Culture

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