Enriques surface

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: Marked Enriques surfaces form a connected 10-dimensional family, which showed is rational. ==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. (The term "singular" does not mean that the surface has singularities, but means that the surface is "special" in some way.) 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==

• A Reye congruence is the family of lines contained in at least 2 quadrics of a given 3-dimensional linear system of quadrics in P3. If the linear system is generic then the Reye congruence is an Enriques surface. These were found by , and may be the earliest 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 ::w^2x^2y^2 + w^2x^2z^2 + w^2y^2z^2 + x^2y^2z^2 + 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 family of examples found by . • 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. For example, if S is the K3 surface w4 + x4 + y4 + z4 = 0 and T is the order 4 automorphism taking (w,x,y,z) to (w,ix,–y,–iz) then T2 has eight fixed points. Blowing up these eight points and taking the quotient by T2 gives a K3 surface with a fixed-point-free involution T, and the quotient of this by T is an Enriques surface. Alternatively, the Enriques surface can be constructed by taking the quotient of the original surface by the order 4 automorphism T and resolving the eight singular points of the quotient. Another example is given by taking the intersection of 3 quadrics of the form Pi(u,v,w) + Qi(x,y,z) = 0 and taking the quotient by the involution taking (u:v:w:x:y:z) to (–x:–y:–z:u:v:w). For generic quadrics this involution is a fixed-point-free involution of a K3 surface so the quotient is an Enriques surface. ==See also==