Enriques surface
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In mathematics, an Enriques surface is an algebraic surface such 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.
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[edit] Invariants
The plurigenera Pn 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 II1,9 of dimension 10 and signature -8 and a group of order 2.
Hodge diamond:
| 1 | ||||
|---|---|---|---|---|
| 0 | 0 | |||
| 0 | 10 | 0 | ||
| 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.
[edit] 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.
[edit] See also
[edit] 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
