Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?

Question

Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.

Consider the condition $$\omega_C\simeq N_{C/S}$$ For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and the above condition is satisfied if and only if the class of $C$ is divisible by $2$ in $\mathrm{Pic}(S)$ (i.e. the exceptions are the two half-pencils of the elliptic fibration).

When $g>1$, do we have smooth curves $C$ for which the above condition holds?

I am especially interested in the case when $S$ is a generic Enriques.

Answer 1

The adjunction formula gives $\omega _C\cong N_{S/C}\otimes \omega _{S|C}$, so your condition is equivalent to say that the canonical class of $S$ restricts to $0$ on $C$. This will never happen if $g\geq 2$. Indeed it means that the universal double covering $\pi :\tilde{S}\rightarrow S $ becomes trivial above $C$. This is impossible if $C^2>0$ because each component of $\pi ^{-1}(C)$ should have positive square, contradicting Hodge index theorem.