Enriques surface
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In mathematics, Enriques surfaces, discovered by Enriques (1949), are complex algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K 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.
Enriques surfaces can also be defined over other fields. Over fields of characteristic other than 2, Michael (Artin 1960) showed that the theory is similar to that over the complex numbers. Over fields of characteristic 2 the definition is modified, and there are two new families, called singular and supersingular Enriques surfaces, described by (Bombieri & Mumford 1976).
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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.
[edit] Characteristic 2
In characteristic 2 there are some new families of Enriques surfaces, sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces. In characteristc 2 the definition of Enriques surfaces is modified: they are defined to be minimal surfaces whose canonical class K is numerically equivalent to 0 and whose second Betti number is 10. (In characteristics other than 2 this is equivalent to the usual definition.) There are now 3 families of Enriques surfaces:
- Classical: dim(H1(O)) = 0. This imples 2K=0 but K is nonzero, and Picτ is Z/2Z. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme μ2.
- Singular: dim(H1(O)) = 1 and is acted on non-trivially by the Frobenius endomorphism. This implies K=0, and Picτ is μ2. The surface is a quotient of a K3 surface by the group scheme Z/2Z.
- Supersingular: dim(H1(O)) = 1 and is acted on trivially by the Frobenius endomorphism. This implies K=0, and Picτ is α2. The surface is a quotient of a reduced singular Gorenstein surface by the group scheme α2.
All Enriques surfaces are elliptic or quasi elliptic.
[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, such as
- w2x2y2 + w2x2z2 + w2y2z2 + x2y2z2 + wxyzQ(w,x,y,z) = 0
- for some general homogeneous polynomial Q of degree 2. 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 in characteristic other than 2 can be constructed like this.
[edit] See also
[edit] References
- Artin, Michael (1960), On Enriques surfaces, PhD thesis, Harvard
- 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.
- Bombieri, Enrico; Mumford, David (1976), "Enriques' classification of surfaces in char. p. III.", Inventiones Mathematicae 35: 197–232, doi:, MR0491720, ISSN 0020-9910
- Cossec, François R.; Dolgachev, Igor V. (1989), Enriques surfaces. I, Progress in Mathematics, 76, Boston, MA: Birkhäuser Boston, MR986969, ISBN 978-0-8176-3417-9
- Enriques, Federigo (1949), Le Superficie Algebriche, Nicola Zanichelli, Bologna, MR0031770, http://www.math.biu.ac.il/~leyenson/library.classical-algebraic-geometry/self-scanned/enriques/enriques.le-superficie-algebriche.1949.300dpi.djvu
