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Given a dominant morphism $\varphi\colon C\to C'$ of curves, a nonsingular point $Q\in C'$, such that $\varphi^{-1}(Q) = \{P_1,\ldots, P_m\}$ consists of nonsingular points only.

Then it is clear to me, that $\mathcal{O}_{C',Q}$ is contained in all $\mathcal{O}_{C,P_i}$.

My question is kind of a converse:

If $\mathcal{O}_{C',Q}\subset A\subset k(C)$ and $A$ is a DVR, is it then true, that $A$ is of the form $\mathcal{O}_{C,P_i}$, for some $P_i$?

If not, can one make it true with some additional conditions?

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