Kodaira dimension
Every surface of Kodaira dimension 1 is an elliptic surface (or a quasielliptic surface in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira dimension formula_5, 0, or 1. All Enriques surfaces, all hyperelliptic surfaces, all Kodaira surfaces, some K3 surfaces, some abelian surfaces, and some rational surfaces are elliptic surfaces, and these examples have Kodaira dimension less than 1. An elliptic surface whose base curve "B" is of genus at least 2 always has Kodaira dimension 1, but the Kodaira dimension can be 1 also for some elliptic surfaces with "B" of genus 0 or 1.