Enriques surface
Encyclopedia
In mathematics
, Enriques surfaces, discovered by , are complex algebraic surface
s
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 surface
s of genus 0.
They are quotients of K3 surface
s 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 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 .
II1,9 of dimension 10 and signature -8 and a group of order 2.
Hodge diamond:
Marked Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.
sometimes called quasi Enriques surfaces or non-classical Enriques surfaces or (super)singular Enriques surfaces.
In characteristic 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:
All Enriques surfaces are elliptic or quasi elliptic.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Enriques surfaces, discovered by , are complex algebraic surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s
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 surface
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
s of genus 0.
They are quotients of K3 surface
K3 surface
In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
s 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 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 .
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 latticeUnimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
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.
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 characteristic 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 implies 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.
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
- 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.