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

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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 '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. –  user13559 Sep 11 '11 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 '11 at 20:03
Dear Elkies.Thanks. –  user13559 Sep 11 '11 at 20:22

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