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[B3]B3 R. Borcherds, *An automorphic form related to cubic surfaces*, unpublished manuscript, math.AG/0002079.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