Borcherds’ Method for Enriques Surfaces

Around real Enriques surfaces

We present a brief overview of the classiication of real Enriques surfaces completed recently and make an attempt to systemize the known clas-siication results for other special types of surfaces. Emphasis is also given to the particular tools used and to the general phenomena discovered; in particular , we prove two new congruence type prohibitions on the Euler characteristic of the real part ...

The Period Lattice for Enriques Surfaces

It is well-known that the isomorphism classes of complex Enriques surfaces are in 1-1 correspondence with a Zariski-open subset (D ? H)=? of the quotient of the Hermitian symmetric space D for O(2; 10). Here H is a totally geodesic divisor in D and ? is a certain arithmetic group. In the usual formulation of this result 5], ? is described as the isometry group of a certain integral lattice N of...

Cubic Surfaces and Borcherds Products

Enriques surfaces covered by Jacobian Kummer surfaces

This paper classifies Enriques surfaces whose K3-cover is a fixed Picard-general Jacobian Kummer surface. There are exactly 31 such surfaces. We describe the free involutions which give these Enriques surfaces explicitly. As a biproduct, we show that Aut(X) is generated by elements of order 2, which is an improvement of the theorem of S. Kondo.

Empty Real Enriques Surfaces and Enriques-einstein-hitchin 4-manifolds

N. Hitchin [H] proved that the Euler characteristic χ(E) and signature σ(E) of a compact orientable 4-dimensional Einstein manifold E satisfy the inequality |σ(E)| 6 2 3χ(E), the equality holding only if either E is flat or the universal covering X of E is a K3-surface and π1(E) = 1, Z/2, or Z/2× Z/2. In the latter cases, E is a K3-surface if π1 = 1, an Enriques surface if π1 = Z/2, or the quot...