In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a minimal surface whose Albanese morphism is an elliptic fibration without singular fibres. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.
Invariants The Kodaira dimension is 0.
Hodge diamond: 1 1 1 0 2 0 1 1 1
Classification Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Λ and F are elliptic curves, and G is a subgroup of F (acting on F by translations), which acts on E not only by translations. There are seven families of hyperelliptic surfaces as in the following table. order of KΛGAction of G on E2AnyZ/2Ze → −e2AnyZ/2Z ⊕ Z/2Ze → −e, e → e+c, −c=c3Z ⊕ ZωZ/3Ze → ωe3Z ⊕ ZωZ/3Z ⊕ Z/3Ze → ωe, e → e+c, ωc=c4Z ⊕ Zi;Z/4Ze → ie4Z ⊕ ZiZ/4Z ⊕ Z/2Ze → ie, e → e+c, ic=c6Z ⊕ ZωZ/6Ze → −ωe Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1.
Quasi hyperelliptic surfaces
A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp. They only exist in characteristics 2 or 3. Their second Betti number is 2, the second Chern number vanishes, and the holomorphic Euler characteristic vanishes. They were classified by , who found six cases in characteristic 3 (in which case 6K= 0) and eight in characteristic 2 (in which case 6K or 4K vanishes). Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).
References - the standard reference book for compact complex surfaces