The moduli space of Enriques surfaces and Borcherds products
Author:
Shigeyuki Kondō
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.
[A]A D. Allcock, The period lattice for Enriques surfaces, Math. Ann., 317(2000), 483–488.
[AF]AF D. Allcock, E. Freitag, Cubic surfaces and Borcherds product, math.AG/0002066.
[BB]BB W.L. Baily, Jr., A. Borel, A compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84(1966), 422–528.
[Ba]Ba H.F. Baker, Principles of geometry II, Cambridge University Press 1922.
[BP]BP W. Barth, C. Peters, Automorphisms of Enriques surfaces, Invent. Math., 73 (1983), 383–411.
[B1]B1 R. Borcherds, The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996), 699–710.
[B2]B2 R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math., 132 (1998), 491–562.
[B3]B3 R. Borcherds, An automorphic form related to cubic surfaces, unpublished manuscript, math.AG/0002079.
[C]C J.H. Conway et al., Atlas of Finite Groups, Oxford Univ., Oxford 1985.
[Di]Di J. Dieudonné, La géométrie des groupes classiques (2nd ed.), Springer 1963.
[Do]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]F E. Freitag, Some modular forms related to cubic surfaces, preprint, August 1999.
[FH]FH E. Freitag, C.F. Hermann, Some modular varieties of low dimension, Adv. Math., 152 (2000), 203–287.
[Hi]Hi D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313–373.
[Ho]Ho E. Horikawa, On the periods of Enriques surfaces I, II, Math. Ann., 234 (1978), 78–108, ibid 235 (1978), 217–246.
[I]I J. Igusa, On the graded ring of theta-constants, Amer. J. Math., 86 (1964), 219–246.
[K1]K1 S. Kondō, Enriques surfaces with finite automorphism groups, Japan. J. Math., 12 (1986), 191–282.
[K2]K2 S. Kondō, The rationality of the moduli space of Enriques surfaces, Compositio Math., 91 (1994), 159–173.
[MN]MN S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on cohomology groups, Invent. Math., 77 (1984), 383–397.
[Na]Na Y. Namikawa, Periods of Enriques surfaces, Math. Ann., 270 (1985), 201–222.
[N1]N1 V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv., 14 (1980), 103–167.
[N2]N2 V.V. Nikulin, On a description of the automorphism groups of Enriques surfaces, Soviet Math. Dokl., 30 (1984), 282–285.
[S]S H. Sterk, Compactifications of the period space of Enriques surfaces I, II, Math. Z., 207 (1991), 1–36, ibid 220 (1995), 427–444.
[A]A D. Allcock, The period lattice for Enriques surfaces, Math. Ann., 317(2000), 483–488.
[AF]AF D. Allcock, E. Freitag, Cubic surfaces and Borcherds product, math.AG/0002066.
[BB]BB W.L. Baily, Jr., A. Borel, A compactification of arithmetic quotients of bounded symmetric domains, Ann. Math., 84(1966), 422–528.
[Ba]Ba H.F. Baker, Principles of geometry II, Cambridge University Press 1922.
[BP]BP W. Barth, C. Peters, Automorphisms of Enriques surfaces, Invent. Math., 73 (1983), 383–411.
[B1]B1 R. Borcherds, The moduli space of Enriques surfaces and the fake monster Lie superalgebra, Topology 35 (1996), 699–710.
[B2]B2 R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math., 132 (1998), 491–562.
[B3]B3 R. Borcherds, An automorphic form related to cubic surfaces, unpublished manuscript, math.AG/0002079.
[C]C J.H. Conway et al., Atlas of Finite Groups, Oxford Univ., Oxford 1985.
[Di]Di J. Dieudonné, La géométrie des groupes classiques (2nd ed.), Springer 1963.
[Do]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]F E. Freitag, Some modular forms related to cubic surfaces, preprint, August 1999.
[FH]FH E. Freitag, C.F. Hermann, Some modular varieties of low dimension, Adv. Math., 152 (2000), 203–287.
[Hi]Hi D. Hilbert, Über die vollen Invariantensysteme, Math. Ann., 42 (1893), 313–373.
[Ho]Ho E. Horikawa, On the periods of Enriques surfaces I, II, Math. Ann., 234 (1978), 78–108, ibid 235 (1978), 217–246.
[I]I J. Igusa, On the graded ring of theta-constants, Amer. J. Math., 86 (1964), 219–246.
[K1]K1 S. Kondō, Enriques surfaces with finite automorphism groups, Japan. J. Math., 12 (1986), 191–282.
[K2]K2 S. Kondō, The rationality of the moduli space of Enriques surfaces, Compositio Math., 91 (1994), 159–173.
[MN]MN S. Mukai, Y. Namikawa, Automorphisms of Enriques surfaces which act trivially on cohomology groups, Invent. Math., 77 (1984), 383–397.
[Na]Na Y. Namikawa, Periods of Enriques surfaces, Math. Ann., 270 (1985), 201–222.
[N1]N1 V.V. Nikulin, Integral symmetric bilinear forms and some of their applications, Math. USSR Izv., 14 (1980), 103–167.
[N2]N2 V.V. Nikulin, On a description of the automorphism groups of Enriques surfaces, Soviet Math. Dokl., 30 (1984), 282–285.
[S]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 Kondō
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602 Japan
Email:
kondo@math.nagoya-u.ac.jp
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
