Hilbert modular surface

From Wikipedia, the free encyclopedia

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Hilbert modular surfaces were first described by (Blumenthal 1903, 1904) using some unpublished notes written by Hilbert about 10 years before.

Contents 1 Definitions 2 Singularities 3 Classification of surfaces 4 Examples 5 References 6 External links

[edit] Definitions

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.

[edit] Singularities

Hirzebruch (1953) showed how to resolve the quotient singularities, and (Hirzebruch 1971) showed how to resolve their cusp singularities.

[edit] Classification of surfaces

The papers (Hirzebruch 1971), (Hirzebruch & Van der Ven 1974) and (Hirzebruch & Zagier 1977) identified their type in the classification of algebraic surfaces.

[edit] Examples

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

[edit] References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, MR2030225, ISBN 978-3-540-00832-3 
• Blumenthal, Otto (1903), "Über Modulfunktionen von mehreren Veränderlichen", Mathematische Annalen (Berlin, New York: Springer-Verlag) 56 (4): 509–548, doi:10.1007/BF01444306 
• Blumenthal, Otto (1904), "Über Modulfunktionen von mehreren Veränderlichen", Mathematische Annalen (Berlin, New York: Springer-Verlag) 58 (4): 497–527, doi:10.1007/BF01449486 
• Hirzebruch, Friedrich (1953), "Über vierdimensionale RIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen", Mathematische Annalen 126: 1–22, doi:10.1007/BF01343146, MR0062842, ISSN 0025-5831 
• Hirzebruch, Friedrich (1971), "The Hilbert modular group, resolution of the singularities at the cusps and related problems", Séminaire Bourbaki, 23ème année (1970/1971), Exp. No. 396, Lecture Notes in Math, 244, Berlin, New York: Springer-Verlag, pp. 275–288, doi:10.1007/BFb0058707, MR0417187, ISBN 978-3-540-05720-8 
• Hirzebruch, Friedrich; Van de Ven, Antonius (1974), "Hilbert modular surfaces and the classification of algebraic surfaces", Inventiones Mathematicae 23: 1–29, doi:10.1007/BF01405200, MR0364262, ISSN 0020-9910 
• Hirzebruch, Friedrich; Zagier, Don (1977), "Classification of Hilbert modular surfaces", in Baily, W. L.; Shioda., T., Complex analysis and algebraic geometry, Tokyo: Iwanami Shoten, pp. 43–77, MR0480356, ISBN 978-0-521-09334-7 
• van der Geer, Gerard (1988), Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 16, Berlin, New York: Springer-Verlag, MR930101, ISBN 978-3-540-17601-5 

[edit] External links

• Ehlen, S., A short introduction to Hilbert modular surfaces and Hirzebruach-Zagier cycles, http://www.mathematik.tu-darmstadt.de/~ehlen/files/hilbert_hz.pdf