Enriques surface

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:

Marked Enriques surfaces form a connected 10-dimensional family, which Kondo (1994) showed is rational.

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 Pic^τ 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 Pic^τ 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 Pic^τ 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.

• 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 Reye (1882), 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

${\displaystyle w^{2}x^{2}y^{2}+w^{2}x^{2}z^{2}+w^{2}y^{2}z^{2}+x^{2}y^{2}z^{2}+wxyzQ(w,x,y,z)=0}$

for some general homogeneous polynomial Q of degree 2. Then its normalization is an Enriques surface. This is the family of examples found by Enriques (1896).

• 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 P_i(u,v,w) + Q_i(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.

• List of algebraic surfaces
• Enriques–Kodaira classification
• Supersingular variety

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