selfintersection of curves inside

What are all the possibilities of the self intersection number of a smooth curve inside an Enriques surface?

flag	asked Sep 11 2011 at 19:20 Ru 217●6

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. Ottem Sep 11 2011 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. – Ru Sep 11 2011 at 19:54

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. Elkies Sep 11 2011 at 20:03

Dear Elkies.Thanks. – Ru Sep 11 2011 at 20:22