Enriques surface

In mathematics, an Enriques surface is an algebraic surfacesuch that the irregularity "q" = 0 and the canonical line bundle is non-trivial but has trivial square. Enriques surfaces are all algebraic (and therefore Kähler) and are elliptic surfaces of genus 1.They are quotients of K3 surfaces by a group of order 2 acting without fixed points and their theory is similar to that of algebraic K3 surfaces.

Invariants

The plurigenera "P"_"n" are 1 if "n" is even and 0 if "n" is odd. The fundamental group has order 2. The second cohomology group H²("X", Z) is isomorphic to the sum of the unique even unimodular lattice II_1,9 of dimension 10 and signature -8 and a group of order 2.

Hodge diamond:

		1
	0		0
1		10		1
	0		0
		1

Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.

In characteristic 2 there are some new families of Enriques surfaces,called quasi Enriques surfaces or non-classical Enriques surfaces.

Examples

There seem to be no really easy examples of Enriques surfaces.

* Take a surface of degree 6 in 3 dimensional projective space with double lines along the edges of a tetrahedron. Then its normalization is an Enriques surface. This is the original family of examples found by Enriques.

* The quotient of a K3 surface by a fixed point free involution is an Enriques surface, and all Enriques surfaces can be constructed like this.

ee also

*list of algebraic surfaces
*Enriques-Kodaira classification
*K3 surface
*elliptic surface

References

*"Compact Complex Surfaces" by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven ISBN 3-540-00832-2 This is the standard reference book for compact complex surfaces.

*"Enriques Surfaces" by F. Cossec, Dolgachev ISBN 0-8176-3417-7

External links

* [http://enriques.mathematik.uni-mainz.de/docs/enriques.shtml Enriques surfaces]