Enriques surface

Summary

An Enriques surface is a smooth projective algebraic surface with no holomorphic $1$-forms and no holomorphic $2$-forms. Its canonical divisor is not trivial, but twice it is trivial. Over $\C$, every Enriques surface is the quotient of a K3 surface by an involution without fixed points. Enriques surfaces form one of the four classes of minimal surfaces of Kodaira dimension $0$. They are non-rational surfaces with $q = p_g = 0$, which shows that Castelnuovo's rationality criterion needs the second plurigenus.