Hilbert modular surface

# Hilbert modular surface

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In mathematics, a Hilbert modular surface is one of the surfaces obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group.

Hilbert modular surfaces were first described by using some unpublished notes written by Hilbert about 10 years before.

## Definitions

If R is the ring of integers of a real quadratic field, then
the Hilbert modular group SL2(R) acts on the product H×H of two copies of the upper half plane H.
There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces:
• The surface X is the quotient of H×H by SL2(R); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.
• The surface X* is obtained from X by adding a finite number of points corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of X, but also singularities at its cusps.
• The surface Y is obtained from .X* by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal.
• The surface Y0 is obtained from Y by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.

There are several variations of this construction:
• The Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup.
• One can extend the Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane.

## Singularities

showed how to resolve the quotient singularities, and showed how to resolve their cusp singularities.

## Classification of surfaces

The papers , and identified their type in the classification of algebraic surfaces. Most of them are surfaces of general type, but several are rational surface
Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...

s or blown up K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....

s or elliptic surface
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....

s.

## Examples

gives a long table of examples.

The Clebsch surface
Clebsch surface
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface is a cubic surface studied by and all of whose 27 exceptional linescan be defined over the real numbers...

blown up at its 10 Eckardt points is a Hilbert modular surface.