Hyperelliptic surface

The Kodaira dimension is 0. Hodge diamond: ==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. 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==