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.

1 Invariants
2 Examples
3 See also
4 References
5 External links

[edit] 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
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.