The adjunction formula for an irreducible curve $D$ on a Enriques surface reads $p_a(D)=\frac{D^2}2+1$. Hence you can probably show that $D^2$ can take every even number $\ge -2$.
– J.C. OttemSep 11 '11 at 19:36

@Ottem: Thanks. Please correct me if I am wrong. Adjunction formula says $K_D=K_X.D \|D + D.D$, if $K_X$ is trivial then your formula is correct. But for Enrique surface $K_X$ is not trivial.
– user13559Sep 11 '11 at 19:54

2

For an Enriques surface $K_X$ is indeed not trivial, but $2K_X \sim 0$, and that's good enough: $K_X\cdot D$ takes integer values, so $2 (K_X \cdot D) = (2K_X \cdot D) = 0$ implies $K_X \cdot D = 0$.
– Noam D. ElkiesSep 11 '11 at 20:03

Dear Elkies.Thanks.
– user13559Sep 11 '11 at 20:22