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The Period Lattice for Enriques Surfaces Daniel Allcock*

Summary: The Period Lattice for Enriques Surfaces
Daniel Allcock*
14 May 1999
MSC: 14J28 (11F55, 11E12)
It is well­known that the isomorphism classes of complex Enriques surfaces are in 1­1 correspon­
dence with a Zariski­open subset (D \Gamma H)=\Gamma of the quotient of the Hermitian symmetric space D
for O(2; 10). Here H is a totally geodesic divisor in D and \Gamma is a certain arithmetic group. In the
usual formulation of this result [5], \Gamma is described as the isometry group of a certain integral lattice
N of signature (2; 10). This lattice is quite complicated, and requires sophisticated techniques to
work with. The purpose of this note is to replace N by the much simpler lattice I 2;10 , the unique
odd unimodular lattice of signature (2; 10). This allows for dramatic simplifications in several
arguments concerning N , replacing intricate analysis by elementary facts. For example, in this
setting it is easy to see that H=\Gamma ` D=\Gamma is irreducible, and also easy to enumerate the boundary
components in the Satake compactification of D=\Gamma. Using I 2;10 in place of N also allows one to
show that (D \Gamma H)=\Gamma has contractible universal cover. The last of these results is new, and the full
proof appears in [1]; here we only give the main idea. The basis of this paper is a lattice­theoretic
trick which is well­known to those who work with lattices; however, its applications in this setting
do not appear to have been published before.
We review some notation and facts from [5]. U denotes the two­dimensional lattice with


Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin


Collections: Mathematics