Алгебралық беттер және Гильберт модулдық кеңістіктер

Algebraic surface in mathematics
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before.

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

Hilbert modular varieties cannot be anabelian.

The papers , and identified their type in the classification of algebraic surfaces. Most of them are surfaces of general type, but several are rational surfaces or blown up K3 surfaces or elliptic surfaces.

gives a long table of examples. The Clebsch surface blown up at its 10 Eckardt points is a Hilbert modular surface.

Given a quadratic field extension for there is an associated Hilbert modular variety obtained from compactifying a certain quotient variety and resolving its singularities. Let denote the upper half plane and let act on viawhere the are the Galois conjugates. The associated quotient variety is denotedand can be compactified to a variety , called the cusps, which are in bijection with the ideal classes in Resolving its singularities gives the variety called the Hilbert modular variety of the field extension. From the Bailey Borel compactification theorem, there is an embedding of this surface into a projective space.