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Suppose that $S$ is a complex algebraic surface of general type.
Let $C$ and $C^{'}$ two distinct smoth curves and $\theta_C$ and $\theta_{C^{'}}$ two theta characteristic on $C$ and $C^{'}$ i.e. $\theta^{\otimes 2}=\omega_C$ same as $C^{'}$ where $\omega_C$ is the canonical line bundle of $C$. I assume that both of the curves have the same theta characteristic i.e. $\theta_C=\theta_{C^{'}}$. I suppose that the genus of both curves is greater than 1. Surely the curves have the same canonical line bundle i.e. $\omega_{C}=\omega_{C^{'}}$. Are there some assumptions that allow me to say that the curves are the same?

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